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Empirical or experimental probabilities of events are based on the outcomes of actual experiments. The probability is constructed on how the event happened during trials. The primary advantage of empirical probability is that the method is free of assumptions. The data is not assumed or has no hypotheses but is supported by experimental examinations and data.Therefore, it is also known as experimental data. Further, it depicts the likelihood of an event happening on the basis of statistical data.
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Key Takeaways: Empirical probability, experiment, frequency, statistical data
Empirical Probability
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Unlike Theoretical probability, Empirical probability is based on the observations of the events that occurred. In other words, empirical probability is the experimental probability that depends on historical data. It is the likelihood of an event based on past recorded data and considers the factual recordings of the events.
For instance, upon rolling a dice four times and observing the corresponding result, the empirical probability of rolling a 5 is 0% (0/4).
Empirical Probability Formula
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The empirical probability P(E) of an event E is given as:
P(E) = Number of times event occurs/Total number of times experiment performed
P(E) = f/n
where,
P(E) is the Empirical Probability
f is the number of trials in which the event occurred, which means the number of times a favorable event occurred.
n is the total number of trials, which means the total amount of time the event was conducted.
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Advantages of Empirical Probability
- The probability is backed by experimentations and actual or statistical data.
Disadvantages of Empirical Probability
- Empirical probability can draw incorrect conclusions. For instance, on tossing a coin, the chances of getting head are half. Nonetheless, tossing a coin several times then getting head in all the tosses is not always correct.
- In small sample sizes, the accuracy of the results gets reduced, and thus large sample sizes are preferred for empirical probability.
Things to Remember
- The empirical or experimental probability of an event is based on what has occurred.
- Empirical probability is defined by the formula
- Empirical probability is based on the past recorded data or statistical data.
- As the number of trials in an experiment increases, the experimental and theoretical probabilities can be expected to become almost the same.
Sample Questions
Ques: A coin toss three times and the result was three heads. By the empirical probability formula, what is the empirical probability of getting a head? (3 Marks)
Ans: Total number of trials = 3
Number of heads = 3
As we know, P(E) = f/n
Therefore, Empirical probability = 3 / 3 = 1.
Hence, the empirical probability of getting a head is 1.
Ques: A dice rolled a thousand times and 5 appeared 140 times. What is the empirical probability of a roll of 5? (3 Marks)
Ans: Total number of trials = 1000
Number of trials in which 5 appears = 140
Thus, empirical probability = P(E) = f/n
140/ 1000 = 7/50.
Ques: A coin is tossed 20 times. Find the empirical probability of the getting head (3 Marks)
| Trial | Frequency |
| Head | 13 |
| Tail | 7 |
| Total | 20 |
Ans: Total number of trials = 20
Number of trails of getting a head = 13
Using, the empirical probability formula, P(E) = f/n
13/20 = 0.65
Ques: In a group of 50 people, 32 people chose to order non-veg noodles over the veg. Find out the empirical probability of someone ordering veg noodles? (3 Marks)
Ans: Total number of people = 50
Number of people who chose non-veg noodles = 32
Number of people who chose veg noodles = 50 - 32 = 18
Therefore, according to the empirical probability formula, it is
P(E) = f/n
18/50 = 0.36.
Hence, the empirical probability of someone ordering veg noodles is 0.36.
Ques: A dice was rolled 120 times. Find the empirical probability of occurrence of 4. (3 Marks)
Ans: Total number of trails = 120
Number of times 4 occured = 1/6 of 120
By empirical probability formula, P(E) = f/n
Therefore, the empirical probability = 20/120 = 1/6 = 0.16.
Ques: A taste test of 100 people revealed that 65 people prefer apples and the remaining prefer bananas. What is the empirical probability that a person prefers apples over bananas? (3 Marks)
Ans: Number of times people preferred apple = 65
Total number of trails = 100
By using empirical probability formula, P(E) = f/n
65/100 = 0.65
Thus, the empirical probability of a person preferring apples over bananas is 0.65.
Ques: Four balls of different colors - yellow, red, green, and blue are kept in a box. One ball is picked each time, and this is done 40 times. The observations are given in the table below. (3 Marks)
| Trial | Frequency |
| Yellow | 12 |
| Red | 15 |
| Green | 6 |
| Blue | 7 |
What is the probability of getting a blue ball when a ball is drawn at random? (2 Marks)
Ans: Number of times blue ball was picked = 7
Total number of trails = 40
P(E) = f/n
7/40 = 0.175
Hence, the empirical probability of getting a blue ball is 0.175.
Ques: 120 people attended a lunch, 80 people preferred mushrooms and others preferred kebabs. Find out the empirical probability of a person to choose mushrooms? (2 Marks)
Ans: Number of times people preferred mushrooms = 80
Total number of people who attended the lunch = 120
P(E)= f/n
80/120 = 2/3 = 0.67
Therefore, the empirical probability of a person to choose mushrooms is 0.67.
Ques: Find out the empirical probability of an event where 40 is the frequency of the class and 120 is the total frequencies in the probability distribution. (2 Marks)
Ans: Frequency of class=40
Total frequency = 120
By the empirical probability formula, we know that
P(E) = f/n
P(E) = 40/120
Hence, P(E) = 0.33333
Ques: In a buffet, 90 out of 100 people chose to order coffee over tea. What is the empirical probability of someone ordering coffee? (3 Marks)
Ans: Total number of people = 100
Number of people who choose coffee over tea = 90.
Number of people who choose tea = 100 - 90 = 10
Thus, as per empirical probability formula,
P(E) =f/n
90/100 = 0.90.
Hence, the empirical probability of someone ordering coffee is 0.90.
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