Example 1. Let’s take an example of tossing a coin, tossing it 40 times , and recording the observations. By using the formula, we can find the experimental probability for heads and tails as shown in the below table.
Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome First H Eleventh T Twenty-first T Thirty-first T Second T Twelfth T Twenty-second H Thirty-second H Third T Thirteenth H Twenty-third T Thirty-third T Fourth H Fourteenth H Twenty-fourth H Thirty-fourth H Fifth H Fifteenth H Twenty-fifth T Thirty-fifth T Sixth H Sixteenth H Twenty-sixth H Thirty-sixth T Seventh T Seventeenth T Twenty-seventh T Thirty-seventh T Eighth H Eighteenth T Twenty-eighth T Thirty-eighth H Ninth T Nineteenth T Twenty-ninth T Thirty-ninth T Tenth H Twentieth T Thirtieth H Fortieth T The formula for experimental probability: P(H) = Number of Heads ÷ Total Number of Trials = 16 ÷ 40 = 0.4 Similarly, P(H) = Number of Tails ÷ Total Number of Trials = 24 ÷ 40 = 0.6 P(H) + P(T) = 0.6 + 0.4 = 1 Note: Repeat this experiment for ‘n’ times and then you will find that the number of times increases, the fraction of experimental probability comes closer to 0.5. Thus if we add P(H) and P(T), we will get 0.6 + 0.4 = 1 which means P(H) and P(T) is the only possible outcomes.
Example 2. A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 30 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.
Experimental Probability = 30/1000 = 0.03 0.03 = (3/100) × 100 = 3% The probability that you will buy a defective phone is 3% ⇒ Number of defective phones next month = 3% × 50000 ⇒ Number of defective phones next month = 0.03 × 50000 ⇒ Number of defective phones next month = 1500
Example 3. There are about 320 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars?
Now the number of people who do not like electric cars is 1000000 – 300000 = 700000 Experimental Probability = 700000/1000000 = 0.7 And, 0.7 = (7/10) × 100 = 70% The probability that someone chose randomly does not like the electric car is 70% The probability that someone like electric cars is 300000/1000000 = 0.3 Let x be the number of people who love electric cars ⇒ x = 0.3 × 320 million ⇒ x = 96 million The number of people who love electric cars is 96 million.
Problem 1: A coin is flipped 200 times, and it lands on heads 120 times. What is the experimental probability of getting heads?
Problem 2: A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of rolling a 3?
Problem 3: In a class survey, 150 students were asked if they prefer reading books or watching movies. 90 students said they prefer watching movies. What is the experimental probability that a randomly chosen student prefers watching movies?
Problem 4: A bag contains 5 red, 7 blue, and 8 green marbles. If 40 marbles are drawn at random with replacement, and 12 of them are red, what is the experimental probability of drawing a red marble?
Problem 5: A basketball player made 45 successful free throws out of 60 attempts. What is the experimental probability that the player will make a free throw?
Problem 6: During a game, a spinner is spun 80 times, landing on a specific section 20 times. What is the experimental probability of the spinner landing on that section?
Define experimental probability..
Probability of an event based on an actual trail in physical world is called experimental probability.
Experimental Probability is calculated using the following formula: P(E) = (Number of trials taken in which event A happened) / Total number of trials
No, experimental probability can’t be used to predict future outcomes as it only achives the theorectical value when the trails becomes infinity.
Theoretical probability is the probability of an event based on mathematical calculations and assumptions, whereas experimental probability is based on actual experiments or trials.
There are some limitation of experimental probability, which are as follows: Experimental probability can be influenced by various factors, such as the sample size, the selection process, and the conditions of the experiment. The number of trials conducted may not be sufficient to establish a reliable pattern, and the results may be subject to random variation. Experimental probability is also limited to the specific conditions of the experiment and may not be applicable in other contexts.
As experimental probability is given by: P(E) = Number of trials taken in which event A happened/Total number of trials Thus, it can’t be negative as both number are count of something and counting numbers are 1, 2, 3, 4, …. and they are never negative.
There are two forms of calculating the probability of an event that are, Theoretical Probability Experimental Probability
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Here we will learn about experimental probability, including using the relative frequency and finding the probability distribution.
There are also probability distribution worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
Experimental probability i s the probability of an event happening based on an experiment or observation.
To calculate the experimental probability of an event, we calculate the relative frequency of the event.
We can also express this as R=\frac{f}{n} where R is the relative frequency, f is the frequency of the event occurring, and n is the number of trials of the experiment.
If we find the relative frequency for all possible events from the experiment we can write the probability distribution for that experiment.
The relative frequency, experimental probability and empirical probability are the same thing and are calculated using the data from random experiments. They also have a key use in real-life problem solving.
For example, Jo made a four-sided spinner out of cardboard and a pencil.
She spun the spinner 50 times. The table shows the number of times the spinner landed on each of the numbers 1 to 4. The final column shows the relative frequency.
The relative frequencies of all possible events will add up to 1.
This is because the events are mutually exclusive.
Step-by-step guide: Mutually exclusive events
You can see that the relative frequencies are not equal to the theoretical probabilities we would expect if the spinner was fair.
If the spinner is fair, the more times an experiment is done the closer the relative frequencies should be to the theoretical probabilities.
In this case the theoretical probability of each section of the spinner would be 0.25, or \frac{1}{4}.
Step-by-step guide: Theoretical probability
In order to calculate an experimental probability distribution:
Draw a table showing the frequency of each outcome in the experiment.
Determine the total number of trials.
Write the experimental probability (relative frequency) of the required outcome(s).
Get your free experimental probability worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Experimental probability is part of our series of lessons to support revision on probability distribution . You may find it helpful to start with the main probability distribution lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
Example 1: finding an experimental probability distribution.
A 3 sided spinner numbered 1,2, and 3 is spun and the results recorded.
Find the probability distribution for the 3 sided spinner from these experimental results.
A table of results has already been provided. We can add an extra column for the relative frequencies.
2 Determine the total number of trials
3 Write the experimental probability (relative frequency) of the required outcome(s).
Divide each frequency by 110 to find the relative frequencies.
A normal 6 sided die is rolled 50 times. A tally chart was used to record the results.
Determine the probability distribution for the 6 sided die. Give your answers as decimals.
Use the tally chart to find the frequencies and add a row for the relative frequencies.
The question stated that the experiment had 50 trials. We can also check that the frequencies add to 50.
Divide each frequency by 50 to find the relative frequencies.
A student made a biased die and wanted to find its probability distribution for use in a game. They rolled the die 100 times and recorded the results.
By calculating the probability distribution for the die, determine the probability of the die landing on a 3 or a 4.
The die was rolled 100 times.
We can find the probability of rolling a 3 or a 4 by adding the relative frequencies for those numbers.
P(3 or 4) = 0.22 + 0.25 = 0.47
A research study asked 1200 people how they commute to work. 640 travelled by car, 174 used the bus, and the rest walked. Determine the relative frequency of someone not commuting to work by car.
Writing the known information into a table, we have
We currently do not know the frequency of people who walked to work. We can calculate this as we know the total frequency.
The number of people who walked to work is equal to
1200-(640+174)=386.
We now have the full table,
The total frequency is 1200.
Divide each frequency by the total number of people (1200), we have
The relative frequency of someone walking to work is 0.321\dot{6} .
In order to calculate a frequency using an experimental probability:
Multiply the total frequency by the experimental probability.
A dice was rolled 300 times. The experimental probability of rolling an even number is \frac{27}{50}. How many times was an even number rolled?
An even number was rolled 162 times.
A bag contains different coloured counters. A counter is selected at random and replaced back into the bag 240 times. The probability distribution of the experiment is given below.
Determine the number of times a blue counter was selected.
As the events are mutually exclusive, the sum of the probabilities must be equal to 1. This means that we can determine the value of x.
1-(0.4+0.25+0.15)=0.2
The experimental probability (relative frequency) of a blue counter is 0.2.
Multiplying the total frequency by 0.1, we have
240 \times 0.2=48.
A blue counter was selected 48 times.
It is common to forget to use the relative frequencies from experiments for probability questions and use the theoretical probabilities instead. For example, they may be asked to find the probability of a die landing on an even number based on an experiment and the student will incorrectly answer it as 0.5.
The relative frequency is the same as the experimental probability. This value is written as a fraction, decimal or percentage, not an integer.
1. A coin is flipped 80 times and the results recorded.
Determine the probability distribution of the coin.
As the number of tosses is 80, dividing the frequencies for the number of heads and the number of tails by 80, we have
2. A 6 sided die is rolled 160 times and the results recorded.
Determine the probability distribution of the die. Write your answers as fractions in their simplest form.
Dividing the frequencies of each number by 160, we get
3. A 3 -sided spinner is spun and the results recorded.
Find the probability distribution of the spinner, giving you answers as decimals to 2 decimal places.
Dividing the frequencies of each colour by 128 and simplifying, we have
4. A 3 -sided spinner is spun and the results recorded.
Find the probability of the spinner not landing on red. Give your answer as a fraction.
Add the frequencies of blue and green and divide by 128.
5. A card is picked at random from a deck and then replaced. This was repeated 4000 times. The probability distribution of the experiment is given below.
How many times was a club picked?
6. Find the missing frequency from the probability distribution.
The total frequency is calculated by dividing the frequency by the relative frequency.
1. A 4 sided spinner was spun in an experiment and the results recorded.
(a) Complete the relative frequency column. Give your answers as decimals.
(b) Find the probability of the spinner landing on a square number.
Total frequency of 80.
2 relative frequencies correct.
All 4 relative frequencies correct 0.225, \ 0.2, \ 0.3375, \ 0.2375.
Relative frequencies of 1 and 4 used.
0.4625 or equivalent
2. A 3 sided spinner was spun and the results recorded.
Complete the table.
Process to find total frequency or use of ratio with 36 and 0.3.
3. Ben flipped a coin 20 times and recorded the results.
(a) Ben says, “the coin must be biased because I got a lot more heads than tails”.
Comment on Ben’s statement.
(b) Fred takes the same coin and flips it another 80 times and records the results.
Use the information to find a probability distribution for the coin.
Stating that Ben’s statement may be false.
Mentioning that 20 times is not enough trials.
Evidence of use of both sets of results from Ben and Fred.
Process of dividing by 100.
P(heads) = 0.48 or equivalent
P(tails) = 0.52 or equivalent
You have now learned how to:
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Home / United States / Math Classes / 7th Grade Math / Experimental and Theoretical Probability
Probability is a branch of math that studies the chance or likelihood of an event occurring. There are two types of prob ability for a particular event: experimental probability and theoretical probability. Learn the difference between the two types of probabilities and the steps involved in their calculation. ...Read More Read Less
Th e chance of a happening is named as the probability of the event happening. It tells us how likely an occasion is going to happen; it doesn’t tell us what’s happening. There is a fair chance of it happening (happening/not happening). They’ll be written as decimals or fractions . The probability of occurrence A is below.
P (A) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of total possible outcomes}}\)
Following are two varieties of probability:
Definition : Probability that’s supported by repeated trials of an experiment is named as experimental probability.
P (event) = \(\frac{\text{Number of times that event occurs}}{\text{Total number of trails}}\)
Example: The table shows the results of spinning a penny 62 times. What’s the probability of spinning heads?
23 | 39 |
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Solution: Heads were spun 23 times in a total of 23 + 39 = 62 spins.
P (heads) = \(\frac{\text{23}}{\text{69}}\) = 0.37 or 37.09 %
Definition : When all possible outcomes are equally likely the theoretical possibility of an incident is that the quotient of the number of favorable outcomes and therefore the number of possible outcomes.
P (event) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\)
Example: You randomly choose one among the letters shown. What’s the theoretical probability of randomly choosing an X?
Solution: P (x) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\) = \(\frac{\text{1}}{\text{7}}\) or 14.28%
A prediction could be a reasonable guess about what is going to happen in the future. Good predictions should be supported by facts and probability.
Predictions supported theoretical probability. These are the foremost reliable varieties of predictions, based on physical relationships that are easy to work and measure which don’t change over time. They include such things as:
Let’s take a look at some differences between experimental and theoretical probability:
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Experimental probability relies on the information which is obtained after an experiment is administered. | Theoretical probability relies on what’s expected to happen in an experiment, without actually conducting it. |
Experimental probability is that the results of the quantity of occurrences of a happening / the whole number of trials | Theoretical probability is that the results of the quantity of favorable outcomes / the entire number of possible outcomes |
A coin is tossed 10 times. It’s recorded that heads occurred 6 times and tails occurred 4 times. P(heads) = \(\frac{6}{10}\) = \(\frac{3}{5}\) P(tails) = \(\frac{4}{10}\) = \(\frac{2}{5}\) | A coin is tossed. P(heads) = \(\frac{1}{2}\)
P(tails) = \(\frac{1}{2}\) |
1. What is the probability of tossing a variety cube and having it come up as a two or a three?
Solution:
First, find the full number of outcomes
Outcomes: 1, 2, 3, 4, 5, and 6
Total outcomes = 6
Next, find the quantity of favorable outcomes.
Favorable outcomes:
Getting a 2 or a 3 = 2 favorable outcomes
Then, find the ratio of favorable outcomes to total outcomes.
P (Event) = Number of favorable outcomes : total number of outcomes
P (2 or 3) = 2:6
P (2 or 3) = 1:3
The solution is 1:3
The theoretical probability of rolling a 2 or a 3 on a variety of cube is 1:3.
2 . A bag contains 25 marbles. You randomly draw a marble from the bag, record its color, so replace it. The table shows the results after 11 draws. Predict the amount of red marbles within the bag.
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Blue | 1 |
Green | 3 |
Red | 5 |
Yellow | 2 |
To seek out the experimental probability of drawing a red marble.
P (EVENT) = \(\frac{\text{Number of times the event occurs}}{\text{Total number of trials}}\)
P (RED) = \(\frac{\text{5}}{\text{11}}\) (You draw red 5 times. You draw a complete of 11 marbles)
To make a prediction, multiply the probability of drawing red by the overall number of marbles within the bag.
\(\frac{\text{5}}{\text{11}}\) x 25 = 11.36 ~ 11 so you’ll be able to predict that there are 11 red balls in an exceedingly bag
3. A spinner was spun 1000 times and the frequency of outcomes was recorded as in the given table.
| Red | Orange | Purple | Yellow | Green |
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| 185 | 195 | 210 | 206 | 204 |
Find (a) list the possible outcomes that you can see in the spinner (b) compare the probability of each outcome (c) find the ratio of each outcome to the total number of times that the spinner spun.
(a) T he possible outcomes are 5. They are red, orange, purple, yellow, and green. Here all the five colors occupy the same area in the spinner. They are all equally likely.
(b) Compute the probability of each event.
P (Red) = \(\frac{\text{Favorable outcomes of red}}{\text{Total number of possible outcomes}}\) = \(\frac{\text{1}}{\text{5}}\) = 0.2
Similarly, P (Orange), P (Purple), P (Yellow) and P (Green) are also \(\frac{\text{1}}{\text{5}}\) or 0.2.
(c) From the experiment the frequency was recorded in the table.
Ratio for red = \(\frac{\text{Number of outcomes of red in the above experiment}}{\text{Number of times the spinner was spun}}\) = \(\frac{\text{185}}{\text{1000}}\) = 0.185
Similarly, we can find the corresponding ratios for orange, purple, yellow, and green are 0.195, 0.210, 0.206, and 0.204 respectively. Can you see that each of the ratios is approximately equal to the probability which we have obtained in (b) [i.e. before conducting the experiment]
The experimental probability of an occurrence is predicted by actual experiments and therefore the recordings of the events. It’s adequate to the amount of times an incident occurred divided by the overall number of trials.
When all possible events or outcomes are equally likely to occur, the theoretical probability is found without collecting data from an experiment.
Experimental probability, also called Empirical probability, relies on actual experiments and adequate recordings of the happening of events. To work out the occurrence of any event, a series of actual experiments are conducted.
Theoretical probability describes how likely an occurrence is to occur. We all know that a coin is equally likely to land heads or tails, therefore the theoretical probability of getting heads is 1/2. Experimental probability describes how frequently a happening actually occurred in an experiment.
So the results of flipping a coin should be somewhere around 50% heads and 50% tails since that’s the theoretical probability.
No, since the quantity of trials during which the event can happen can not be negative and also the total number of trials is usually positive.
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Probability is the study of chances and is an important topic in mathematics. There are two types of probability: theoretical and experimental.
So, how to define theoretical and experimental probability? Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.
Read on to find out the differences between theoretical and experimental probability. If you wonder How to Understand Statistics Easily , I wrote a whole article where I share 9 helpful tips to help you Ace statistics.
Table of Contents
Theoretical probability is calculated using mathematical formulas. In other words, a theoretical probability is a probability that is determined based on reasoning. It does not require any experiments to be conducted. Theoretical probability can be used to calculate the likelihood of an event occurring before it happens.
Keep in mind that theoretical probability doesn’t involve any experiments or surveys; instead, it relies on known information to calculate the chances of something happening.
For example, if you wanted to calculate the probability of flipping a coin and getting tails, you would use the formula for theoretical probability. You know that there are two possible outcomes—heads or tails—and that each outcome is equally likely, so you would calculate the probability as follows: 1/2, or 50%.
The theoretical probability formula is defined as follows: Theoretical Probability = Number of favorable (desirable) outcomes divided by the Number of possible outcomes.
Probability plays a vital role in the day to day life. Here is how theoretical probability is used in real life:
Experimental probability, on the other hand, is based on results from experiments or surveys. It is the ratio of the number of successful trials divided by the total number of trials conducted. Experimental probability can be used to calculate the likelihood of an event occurring after it happens.
For example, if you flipped a coin 20 times and got heads eight times, the experimental probability of obtaining heads would be 8/20, which is the same as 2/5, 0.4, or 40%.
The formula for the experimental probability is as follows: Probability of an Event P(E) = Number of times an event happens divided by the Total Number of trials .
If you are interested in learning how to calculate experimental probability, I encourage you to watch the video below.
Knowing experimental probability in real life provides powerful insights into probability’s nature. Here are a few examples of how experimental probability is used in real life:
The main difference between theoretical and experimental probability is that theoretical probability expresses how likely an event is to occur, while experimental probability characterizes how frequently an event occurs in an experiment.
In general, the theoretical probability is more reliable than experimental because it doesn’t rely on a limited sample size; however, experimental probability can still give you a good idea of the chances of something happening.
The reason is that the theoretical probability of an event will invariably be the same, whereas the experimental probability is typically affected by chance; therefore, it can be different for different experiments.
Also, generally, the more trials you carry out, the more times you flip a coin, and the closer the experimental probability is likely to be to its theoretical probability.
Also, note that theoretical probability is calculated using mathematical formulas, while experimental probability is found by conducting experiments.
Theoretical and experimental probabilities are two ways of calculating the likelihood of an event occurring. Theoretical probability uses mathematical formulas, while experimental probability uses data from experiments. Both types of probability are useful in different situations.
I believe that both theoretical and experimental probabilities are important in mathematics. Theoretical probability uses mathematical formulas to calculate chances, while experimental probability relies on results from experiments or surveys.
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The experimental probability of an event is the proportion of times the event occurs in a given number of trials.
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The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment. More generally, empirical probability estimates probabilities from experience and observation.
When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events , all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis .
Theoretical probability is the likelihood of an event occurring, calculated using theoretical mathematics. Experimental probability is the likelihood of an event occurring, as determined by observation.
Experimental probability is the probability of an event occurring as determined by data from a series of repeated experiments. The probability is determined by counting the number of times the event occurs divided by the total number of trials.
An experimental probability is a probability that is calculated from a set of experiments. This type of probability is used to calculate the chances of something happening, based on the results of past experiments.
The experimental probability of an event is the ratio of the number of times the event occurs to the total number of trials.
The probability of flipping a coin and getting heads is 1/2.
The probability of flipping a coin and getting tails is 1/2.
1. In a jar there are five red balls and three green balls. If you draw a ball at random from the jar, what is the probability that you will draw a red ball?
The probability of drawing a red ball is 5/8.
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Experimental Probability is defined as a branch of mathematics that deals with the uncertainty of the occurrence of events . It deals with the probability of outcomes of an experiment.
P(E) = Number of time an event occur/Total number of time an experiment is performed
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Key Terms: Experimental Probability, Probability, Emperical Probability, Sample Space, Experiment, Theoretical Probability, Statistics, Population, Trials, Set, Statistical Analysis
[Click Here for Sample Questions]
Experimental Probability is the probability of an event based on exact recordings or experiments of an event. The calculation is done by dividing the number of times an event occurred by the total number of trials in an experiment.
Experimental Probability = Number of times a particular event occurs/ Number of total trials
Experimental Probability
A farmer wants to know the probability a new cauliflower seed will sprout. He sows 2,000 seeds and finds that 1820 sprout. The experimental probability is 1820/2000. A coin is tossed 1500 times i.e. the total number of trials is 1500. The event of getting head and tail is mentioned as H and T, respectively. Totally H has happened 550 times. Find the experimental probability of all the outcomes of H and T respectively. Given, Total number of trials = 1500 times Total number of H happened = 550 So, the total number of T happened = Total number of trials - Total number of H happened Total number of T happened = 1500 - 550 = 950 To calculate the probability of occurrence of an event, use the formula as Probability of getting H, P(H) = Total number of H happened/Total number of trials P(H) = 550/1500 P(H) = 11/30 = 0.367 Probability of getting T, P(T) = Total number of T happened/Total number of trials P(T) = 950/1500 P(T) = 19/30 = 0.633 Therefore, P(H) = 0.367, P(T) = 0.633 P(H) + P(T) = 0.367 + 0.633 = 1 The 3 coins are tossed 1000 times simultaneously and we get three tails = 160, two tails = 260, one tails = 320, no tails = 260. Calculate the probability of occurrence of each of the above events. Total number of trials = 1000 Three tails, A = 160 Two tails, B = 260 One tails, C = 320 No tails, D = 260 To calculate the probability of occurrence, P of an event, Probability of getting A, P(A) = 160/1000 = 4/25 = 0.16 Probability of getting B, P(B) = 260/1000 = 13/50 = 0.26 Probability of getting C, P(C) = 320/1000 = 8/25 = 0.32 Probability of getting D, P(D) = 260/1000 = 13/50 = 0.26 Therefore, P(A) = 0.16, P(B) = 0.26, P(C) = 0.32, P(D) = 0.26 P(A) + P(B) + P(C) + P(D) = 0.16 + 0.26 + 0.32 + 0.26 = 1 |
Chapter Related Concepts | ||
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Probability is a branch of mathematics used to express a chance of occurrence of an event in a mathematical expression . The probability is higher means the change of occurrence of an event is also higher.
Probability (P(E)) = Number of favourable outcomes of an event/Total Number of outcomes of an event
The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 3/10.
The possible outcomes - (H, H), (H, T), (T, H), and (T, T). The no. of possible outcomes of both coins landing on heads is 1 So, the probability will be 20%. |
Probability
Theoretical Probability describes the probability of the happening of certain events that are not in an experimental way of an occurrence. It will conduct an ananlysis to determine the ideal situation without any experiment.
Theoretical Probability = Total number of desired outcomes/ Total number of outcomes
An example of this is drawing a red stone out of a bag etc. If a bag contains 6 red and 8 blue balls then what is the probability of picking up a red ball? To calculate the theoretical probability the following formula is used. Theoretical Probability = Number of favorable outcomes / Number of possible outcomes. Number of favorable outcomes = 6 Number of possible outcomes = 6 + 8 = 14 P(red) = 6 / 14 |
Theoretical Probability
Steps to find Experimental Probability of an event are as follows:
Class 10 Maths Concepts | ||
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Ques. Find the probability of getting a number on rolling a six-faced die 500 times. (4 marks)
Ans. To calculate the experimental probability,
P(1) = 80/500 = 4/25 = 0.16
P(2) = 60/500 = 3/25 = 0.12
P(3) = 70/500 = 7/50 = 0.14
P(4) = 84/500 = 42/250 = 0.168
P(5) = 120/500 = 6/25 = 0.24
P(6) = 86/500 = 43/250 = 0.172
Therefore, P(1) = 0.16, P(2) = 0.12, P(3) = 0.14, P(4) = 0.168, P(5) = 0.24, P(6) = 0.172
To verify, P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 0.16 + 0.12 + 0.14 + 0.168 + 0.24 + 0.172 = 1
Ques. Find the probability of occurrence of a number of girls in a family having 2 children, the data’s of the family are given in a table. (4 marks)
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A) Calculate the total number of families
B) Calculate the probability of 2 girls in a family
C) Calculate the probability of 1 girl in a family
D) Calculate the probability of at most 1 girl in a family
Ans. A) To calculate the total number of families, use the number of families data from the table,
Total number of families = 320 + 440 + 340 = 1100
B) To calculate the probability of 2 girls in a family, P(2G)
P(2G) = 320/1100 = 16/55 = 0.291
The probability of families containing 2 girls is 0.291
C) To calculate the probability of 1 girl in a family, P(1G)
P(1G) = 440/1100 = 22/55 = 2/5 = 0.4
The probability of families containing 1 girl is 0.4
D) To calculate the probability of at most 1 girl in a family, P(G)
P(G) = (440/1100) + (340/1100) = (22/55) + (17/55) = 39/55 = 0.709
The probability of at most 1 girl in a family is 0.709
To verify: P(2G) + P(G) = 0.291 + 0.709 = 1
Ques. To know the opinion of students about maths and science a survey is taken with 500 students, the results of the survey are given below (4 marks)
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A) Find the probability of students who like maths and science?
B) Find the probability of students who dislike maths and science?
C) Find the probability of students who neither like nor dislike maths and science?
Ans. Given that, Total number of students is 500
A) To calculate the probability of students who like maths and science, P(L)
P(L) = (240/500) + (190/500) = (12/25) + (19/50) = 0.48 + 0.38 = 0.86
The probability of students who like maths and science is 0.86
B) To calculate the probability of students who dislike maths and science, P(D)
P(D) = (130/500) + (80/500) = (13/50) + (4/25) = 0.26 + 0.16 = 0.42
The probability of students who dislike maths and science is 0.42
C) To calculate the probability of students who neither like nor dislike maths and science, P(N)
Number of students who neither like or dislike maths = Total number of students - (Students who like maths + students who dislike maths) = 500 - (340 + 120) = 40
Number of students who neither like or dislike science = Total number of students - (Students who like science + students who dislike science) = 500 - (190 + 80) = 230
P(N) = (40/500) + (230/500) = (2/25) + (23/50) = 0.08 + 0.46 = 0.54
The probability of students who neither like nor dislike maths and science is 0.54
Ques. Find the probability of getting a number on rolling a six-faced die 800 times. (4 marks)
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P(1) = 180/800 = 9/40 = 0.225
P(2) = 160/800 = 1/5 = 0.2
P(3) = 170/800 = 17/80 = 0.212
P(4) = 86/800 = 43/400 = 0.107
P(5) = 120/800 = 3/20 = 0.15
P(6) = 84/800 = 21/200 = 0.105
Therefore, P(1) = 0.225, P(2) = 0.2, P(3) = 0.212, P(4) = 0.107, P(5) = 0.15, P(6) = 0.172
To verify, P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 0.225 + 0.2 + 0.212 + 0.107 + 0.15 + 0.105 = 1
Ques. The three coins are tossed 5000 times, the outcomes of occurrence of all outcomes are given below in the table. Calculate experimental probability. (4 marks)
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Ans. To calculate the probability of occurrence of all outcomes,
Experimental probability = Total number of desired outcomes / Total number of trails
P(HHH) = 640/5000 = 16/125 = 0.128
P(TTT) = 420/5000 = 21/250 = 0.084
P(HTH) = 500/5000 = 1/10 = 0.1
P(THT) = 740/5000 = 37/250 = 0.148
P(HTT) = 540/5000 = 27/250 = 0.108
P(THH) = 720/5000 = 18/125 = 0.144
P(HTH) = 860/5000 = 43/250 = 0.172
P(HHT) = 580/5000 = 29/250 = 0.116
Therefore, P(HHH) = 0.128, P(TTT) = 0.084, P(HTH) = 0.1, P(THT) = 0.148, P(HTT) = 0.108, P(THH) = 0.144, P(HTH) = 0.172, P(HHT) = 0.116
To verify: P(HHH) + P(TTT) + P(HTH) + P(THT) + P(HTT) + P(THH) + P(HTH) + P(HHT)
= 0.128 + 0.084 + 0.1 + 0.148 + 0.108 + 0.144 + 0.172 + 0.116 = 1
Ques. What is the difference between experimental and theoretical probability. (4 marks)
Ans. The difference between experimental and theoretical probability are as follows:
Experimental Probability | Theoretical Probability |
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Experimental Probability is performed by conducting an experiment. | Theoretical Probability is performed by using mathematical analysis and formulas. |
It is based on real-life experiments. | It is based on assumptions. |
The outcomes obtained from the experiment is less accurate. | The outcomes obtained from the experiment is more accurate. |
Example: Tossing a coin multiple times | Example: Drawing a certain card from set of cards. |
Ques. A manufacturer makes 40,000 cell phones every month. After inspecting 2000 phones, the manufacturer found that 50 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month. (3 marks)
Ans. Experimental Probability = 50/2000 = 0.025
Ques. There are about 300 million people living in the USA. Pretend that a survey of 1 million people revealed that 600,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars. (3 marks)
Ans. Since the number of people who do not like electric cars is 1000000 – 300000 = 400000
Ques. Find the probability of getting a number on rolling a six-faced die 500 times. (4 marks)
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P(1) = 100/500 = 1/5 = 0.20
P(2) = 150/500 = 3/10 = 0.3
P(3) = 200/500 = 2/5 = 0.4
P(4) = 250/500 = 5/10 = 0.5
P(5) = 100/500 = 1/5 = 0.2
P(6) = 150/500 = 3/10 = 0.3
Therefore, P(1) = 0.20, P(2) = 0.3, P(3) = 0.4, P(4) = 0.5, P(5) = 0.2, P(6) = 0.3
Ques. The following set of data shows the number of messages that Anil received recently from 6 of his friends. 4, 2, 2, 1, 6, 8. Based on this, find the probability that Anil will receive less than 2 messages next time. (2 marks)
Ans. Mike has received less than 2 messages from 3 of his friends out of 6.
Therefore, P(<2) = 3/6 = ½
Ques. The following table shows the recording of the outcomes on throwing a 6-sided die 200 times. (3 marks)
1 | 15 |
2 | 18 |
3 | 20 |
4 | 27 |
5 | 13 |
6 | 16 |
Find the experimental probability of: A) Rolling a four; B) Rolling a number less than four; C) Rolling a 2 or 5
Ans. Experimental probability is calculated by the formula: Number of times an event occurs/Total number of trials
A) Rolling a 4: 27/200 = 0.135
B) Rolling a number less than 4: 53/100 = 0.005
C) Rolling a 2 or 5: 31/200 = 0.155
Check-Out:
1. find the sums given below : \(7 + 10\frac 12+ 14 + ....... + 84\) \(34 + 32 + 30 + ....... + 10\) \(–5 + (–8) + (–11) + ....... + (–230)\), 2. which of the following are aps if they form an ap, find the common difference d and write three more terms. (i) 2, 4, 8, 16, . . . . (ii) \(2, \frac{5}{2},3,\frac{7}{2}\) , . . . . (iii) – 1.2, – 3.2, – 5.2, – 7.2, . . . . (iv) – 10, – 6, – 2, 2, . . . (v) 3, \(3 + \sqrt{2} , 3 + 3\sqrt{2} , 3 + 3 \sqrt{2}\) . . . . (vi) 0.2, 0.22, 0.222, 0.2222, . . . . (vii) 0, – 4, – 8, –12, . . . . (viii) \(\frac{-1}{2}, \frac{-1}{2}, \frac{-1}{2}, \frac{-1}{2}\) , . . . . (ix) 1, 3, 9, 27, . . . . (x) a, 2a, 3a, 4a, . . . . (xi) a, \(a^2, a^3, a^4,\) . . . . (xii) \(\sqrt{2}, \sqrt{8} , \sqrt{18} , \sqrt {32}\) . . . . (xiii) \(\sqrt {3}, \sqrt {6}, \sqrt {9} , \sqrt {12}\) . . . . . (xiv) \(1^2 , 3^2 , 5^2 , 7^2\) , . . . . (xv) \(1^2 , 5^2, 7^2, 7^3\) , . . . ., 3. a vessel is in the form of an inverted cone. its height is 8 cm and the radius of its top, which is open, is 5 cm. it is filled with water up to the brim. when lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. find the number of lead shots dropped in the vessel., 4. if 3 cot a = 4, check whether \(\frac{(1-\text{tan}^2 a)}{(1+\text{tan}^2 a)}\) = cos 2 a – sin 2 a or not, 5. in a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord, 6. prove the following identities, where the angles involved are acute angles for which the expressions are defined: \(\frac{(\text{1 + tan² a})}{(\text{1 + cot² a})} = (\frac{\text{1 - tan a }}{\text{ 1 - cot a}})^²= \text{tan² a}\).
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How likely something is to happen.
Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.
When a coin is tossed, there are two possible outcomes:
Heads (H) or Tails (T)
When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .
The probability of any one of them is 1 6
In general:
Probability of an event happening = Number of ways it can happen Total number of outcomes
Number of ways it can happen: 1 (there is only 1 face with a "4" on it)
Total number of outcomes: 6 (there are 6 faces altogether)
So the probability = 1 6
Number of ways it can happen: 4 (there are 4 blues)
Total number of outcomes: 5 (there are 5 marbles in total)
So the probability = 4 5 = 0.8
We can show probability on a Probability Line :
Probability is always between 0 and 1
Probability does not tell us exactly what will happen, it is just a guide
Probability says that heads have a ½ chance, so we can expect 50 Heads .
But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.
Learn more at Probability Index .
Some words have special meaning in Probability:
Experiment : a repeatable procedure with a set of possible results.
We can throw the dice again and again, so it is repeatable.
The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}
Outcome: A possible result.
Trial: A single performance of an experiment.
Trial | Trial | Trial | Trial | |
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Head | ✔ | ✔ | ✔ | |
Tail | ✔ |
Three trials had the outcome "Head", and one trial had the outcome "Tail"
Sample Space: all the possible outcomes of an experiment.
There are 52 cards in a deck (not including Jokers)
So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }
The Sample Space is made up of Sample Points:
Sample Point: just one of the possible outcomes
"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.
There are 6 different sample points in that sample space.
Event: one or more outcomes of an experiment
An event can be just one outcome:
An event can include more than one outcome:
Hey, let's use those words, so you get used to them:
The Sample Space is all possible Outcomes (36 Sample Points):
{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}
The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :
{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}
These are Alex's Results:
Trial | Is it a Double? |
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{3,4} | No |
{5,1} | No |
{2,2} | |
{6,3} | No |
... | ... |
After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?
IMAGES
COMMENTS
The experimental probability of an event is based on the number of times the event has occurred during the experiment and the total number of times the experiment was conducted. Each possible outcome is uncertain and the set of all the possible outcomes is called the sample space. The formula to calculate the experimental probability is: P (E ...
Experimental probability is the probability calculated by repeating an experiment and observing the outcomes. Learn the definition, formula, facts and more!
This brings us to experimental probability and its definition. Experimental probability is the probability determined based on the results from performing the particular experiment.
Experimental probability, also known as Empirical probability, is based on actual experiments and adequate recordings of the happening of events. To determine the occurrence of any event, a series of actual experiments are conducted. Experiments which do not have a fixed result are known as random experiments.
Experimental probability. Experimental probability (EP), also called empirical probability or relative frequency, is probability based on data collected from repeated trials. Experimental probability formula. Let n represent the total number of trials or the number of times an experiment is done. Let p represent the number of times an event ...
Learn its definition, formula, and properties through engaging examples and practice problems, all crafted to make mathematics fun and understandable. Get to grips with this key statistical concept and see how it's applied in real-world scenarios.
Learn more about the definition and application of experimental probability. Explore more with solved examples that are a part of this article.
Learn the definition of experimental probability. Understand the probability formula and practice calculating experimental probability using...
Experimental probability is like the bridge between math and the real world, offering a hands-on approach to understanding likelihood and chance. It's all about observation, data collection, and making sense of the patterns that emerge. Experimental probability takes us beyond the theoretical and into the empirical, providing our children with a richer, fuller understanding of how ...
Experimental probability refers to the probability of an event occurring based on the results of an experiment or series of trials. In other words, it is determined by conducting a physical experiment or series of trials and observing the proportion of times that the event of interest occurs.
Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes. It is determined by conducting an experiment or observing an event multiple times and recording the number of times the event occurs.
Definition 4.1.6 4.1. 6. Experimental Probabilities. P(A) = number of times A occurs number of times the experiment was repeated P ( A) = number of times A occurs number of times the experiment was repeated. For the event of getting a 6, the probability would by 163 1000 = 0.163 163 1000 = 0.163.
Experimental probability, also known as empirical probability, is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability, which predicts outcomes based on known possibilities, experimental probability is derived from real-life experiments and observations.
Free experimental probability GCSE maths revision guide, including step by step examples, exam questions and free worksheet.
The experimental probability of an event is calculated based on experimental data, and theoretical probability is calculated by predicting the possible outcomes of an event.
Probability is the likelihood that an event will happen or not. In other words, it describes the possibility of the occurrence of an event. Amongst the different types of events in probability in mathematics theoretical and experimental probability, we will be focusing on experimental probability distribution, its formulas with examples.
Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.
Learn the definition of experimental probability. Understand the probability formula and practice calculating experimental probability using...
The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment. More generally, empirical probability estimates probabilities from experience and observation.
Experimental Probability is defined as a branch of mathematics that deals with the uncertainty of the occurrence of events. It deals with the probability of outcomes of an experiment.Experimental Probability involves a procedure that can be repeated infinitely.
Relative frequency gives a way to measure the proportion of "successful" outcomes when doing an experimental approach. From the interactive applications above, it appears that the relative frequency does jump around as the experiment is repeated but that the amount of variation decreases as the number of experiments increases. This is known to be true in general and is known as the "Law of ...
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.