# Experimental Probability – Definition, Formula, Examples | Solved Questions on Experimental Probability

The chance of occurrence of an event is called probability. The exact value of probability lies between 0 and 1. The probability is determined on the basis of the results of an experiment called experimental probability. We are giving the definition, formula and examples of experimental probability along with the solved problems in the following sections.

## Definition of Experimental Probability

Experimental probability or empirical probability is the probability that is determined on the basis of a series of experiments. The experiment is conducted to check the possibility of an event occurring or not. A random experiment is done and is repeated multiple times to determine their likelihood and every repetition is called a trial. The experiment can be rolling a die, rotating a spinner or tossing a coin.

In mathematics, the probability of an event is defined as the ratio of the number of times an event occurred to the total number of trials. Let us say if you toss a coin 20 times and record the result whether head or tail. The experimental probability of getting head is calculated as a fraction of the number of recorded heads and the total number of tosses.

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### Experimental Probability Examples

Below provided is an example of experimental probability.

1. The numbers designs created by a painter per day in this week are given as 5, 6, 7, 8, 9, 7

Based on this data, what is the reasonable estimate of the probability that the painter designs less than 7 designs the next day?

P(< 7 designs) = $$\frac { 2 }{ 7 }$$ = 0.285 = 28%

2. The number of cakes prepared by Patrick per day in this week are 4, 7, 6, 9, 5, 9, 5.

Based on this data, what is the reasonable estimate of the probability that Patrick makes less than 6 cakes the next day?

P(6 < cakes) = 3/7 = 0.428 = 42%

### Experimental Probability Formula

The experimental probability of an event depends 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 possible outcomes is sample space. The formula to calculate the experimental probability is given here.

Probability of an Event P(E) = Number of times an event occurs / Total number of trials.

### Difference Between Experimental Probability & Theoretical Probability

The differences between the theoretical probability and experimental probability are listed here.

Experimental Probability Theoretical Probability
It is based on the data which is obtained after an experiment is carried out. It is based on what is expected to happen in an experiment, without actually conducting it.
The formula is the number of occurrences of an event ÷ the total number of trials The formula is the number of favourable outcomes ÷ the total number of possible outcomes
Example: A coin is tossed 15 times. It is recorded that heads occurred 8 times and tails occurred 7 times.
P(tail) = 7/15
Example: A coin is tossed.
P(tail) = 1/2

### Solved Problems on Experimental Probability

Problem 1:
The following table shows the recording of the outcomes on throwing a 6 sided die 50 times.

Outcome Frequency
1 8
2 11
3 12
4 10
5 5
6 4

Find the experimental probability of a) Rolling a three, b) Rolling a number less than three, c) Rolling 2 or 4

Solution:
Experimental probability is calculated by the formula: Number of times an event occurs/Total number of trials
a) Rolling a 3: 12/50 = 0.24
b) Rolling a number less than 3: 19/50 = 0.38
c) Rolling a 2 or 4: 21/50 = 0.42

Problem 2:
The following set of data shows the number of messages that David received recently from 6 of his friends. 4, 3, 2, 1, 6, 8. Based on this data, find the probability that David will receive less than 5 messages next time.

Solution:
David received less than 5 messages from 10 messages from 4 of his friends out of 6.
Therefore, P(<5) = 4/6 = 2/3