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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.
- Experimental Probability involves a procedure that can be repeated infinitely.
- In probability, it is also known as trial or empirical probability.
- It works on a well-defined set of outcomes known as sample space.
- A trial is said to be deterministic if it consists of one sample space.
- An experiment is said to be random if it consists of more than one sample space.
- The results of the experiment are used in the field of statistical analysis.
- We can observe a chance of occurrence of a particular event in an experiment.
- Empirical probability helps in calculating the chance of occurrence of a head and tails.
- The formula to calculate the experimental probability is as follows:
P(E) = Number of time an event occur/Total number of time an experiment is performed
- For example, on tossing two coins for six times, there is a chance of getting both heads or tails or pairs of heads and tails each time of the throw.
Key Terms: Experimental Probability, Probability, Emperical Probability, Sample Space, Experiment, Theoretical Probability, Statistics, Population, Trials, Set, Statistical Analysis
What is Experimental Probability?
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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 performs a series of trials to check the occurrence of an event.
- The outcome of each event is uncertain in nature.
- Events that produce uncertain values are called random experiments.
- It can be expressed mathematically as:
Experimental Probability = Number of times a particular event occurs/ Number of total trials
Experimental Probability
Examples of Experimental ProbabilityExample 1: 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. Example 2: 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. Solution: 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 To verify: P(H) + P(T) = 0.367 + 0.633 = 1 Example 3: 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. Solution: 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 To verify: P(A) + P(B) + P(C) + P(D) = 0.16 + 0.26 + 0.32 + 0.26 = 1 |
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| Chapter Related Concepts | ||
|---|---|---|
| Double Integral | Geometric Probability | Conditional probability |
| Geometric Mean | Sum of Probabilities | Rolle’s Theorem |
What is Probability?
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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 describes the likelihood of an event in terms of numeric values.
- The value of an event likes to lie between zero and one.
- It is obtained by the ratio of favourable outcomes to total number of outcomes.
- The different events can be organized with the help of probability tree diagram.
- Some real-life examples of probability includes a batting average in cricket, weather forecasting, flipping a coin or dice etc.
- It can be expressed as:
Probability (P(E)) = Number of favourable outcomes of an event/Total Number of outcomes of an event
Example of What is Probability?Example 1: There are 10 pillows in a bed, 4 are red, 3 are yellow and 3 is blue. What is the probability of picking a yellow pillow? Solution: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 3/10. Example 2: Two coins are flipped 25 times simultaneously. What is the probability of both coins landing on heads? Solution: 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
What is Theoretical Probability?
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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 gives the result of an event based on mathematics and reasoning.
- Prior knowledge is required before calculating the probability of an event.
- This is related to theory of the required probability.
- We can express theoretical probability mathematically as,
Theoretical Probability = Total number of desired outcomes/ Total number of outcomes
Example of What is Theoretical Probability?Example 1: An example of this is drawing a red stone out of a bag etc. Example 2: If a bag contains 6 red and 8 blue balls then what is the probability of picking up a red ball? Solution: 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
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Steps to find Experimental Probability of an event are as follows:
- Conduct an experiment like tossing a coin 100 times or rolling a die 1000 times.
- Note down the number of particular event occurrences and a total number of trials done.
- Divide the two recordings of the experiment.
- It will result in the experimental probability.
- The sum of probability of all occurred events is always one.
- This we can easily verify our calculated probability.
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Things to Remember
- Experimental probability is a calculation of the chance of occurrence of an event of experiment.
- Theoretical probability means calculation of the desired outcome of an event
- The total probability of an event is always 1
- The probability of any event will be between 0 and 1, where 1 denotes a certain event and 0 denotes a non-occurrence of an event.
- Individuals can try Probability Important Questions for further practice regarding the topic.
Sample Questions
Ques. Find the probability of getting a number on rolling a six-faced die 500 times.(4 marks)
| Event | 1 | 2 | 3 | 4 | 5 | 6 |
| Number | 80 | 60 | 70 | 84 | 120 | 86 |
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)
| Girls in a family | 2 | 1 | 0 |
| Number of families | 320 | 440 | 340 |
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)
| Subject | Maths | Science |
|---|---|---|
| Likes | 240 | 190 |
| Dislikes | 130 | 80 |
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)
| Event | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Number | 180 | 160 | 170 | 86 | 120 | 84 |
Ans. To calculate the experimental probability,
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)
| Events | HHH | TTT | HTH | THT | HTT | THH | HTH | HHT |
|---|---|---|---|---|---|---|---|---|
| No of times | 640 | 420 | 500 | 740 | 540 | 720 | 860 | 580 |
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 |
|---|---|
| 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
- 0.025 = (25/100) × 100 = 25%
- The probability that you will buy a defective phone is 25%
- Number of defective phones next month = 25% × 50000
- Number of defective phones next month = 0.025 × 50000
- Number of defective phones next month = 1000
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
- Experimental Probability = 400000/1000000 = 0.4
- 0.4 = (4/10) × 100 = 40%
- The probability that someone chose randomly does not like the electric car is 40%
- The probability that someone like electric cars is 600000/1000000 = 0.6
- Let x be the number of people who love electric cars
- x = 0.6 × 300 million
- x = 18 million
Ques. Find the probability of getting a number on rolling a six-faced die 500 times. (4 marks)
| Event | 1 | 2 | 3 | 4 | 5 | 6 |
| Number | 100 | 150 | 200 | 250 | 100 | 150 |
Ans. To calculate the experimental probability,
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)
| Outcome | Frequency |
|---|---|
| 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
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