Experimental Probability - Formula & Examples

# Experimental Probability

The likeliness of occurrence of something. We have in general two kinds of probabilities: theoretical and experimental. While theoretical probabilities deals with the nature of the experiment and events, experimental probabilities relies on the fact of actual occurrence of the experiment and events.
Experimental probability is an estimate simply for the probabilities that cannot at all be determined by simple logics.

In simple words, experimental probability is the simple ratio of the number of times an event is occurring to the total number of times or trials that the activity has been repeated for.
One can see the use of experimental probabilities in fields of research and social sciences experiments, medicine etc.
The cases where theoretical probabilities are difficult to find we have to rely on experimental probability only. The sample size in case of experimenting probability is generally large or we say that the experiment is repeated for numerous times.

It can be easily noted that as we increase the number of trials the experimental probability approaches more towards the theoretical probability. But for this the number of trials has to be very large.

## Experimental Probability Formula

While theoretical probability is the ratio of the number of favorable outcomes to the total number of outcomes, the experimental probability is the ratio of number of times the event is occurring to the total number of trials of the experiment.
It can be represented mathematically as:

Experimental probability = $\frac{(number\ of\ occurrences\ of\ the\ event)} {(total\ number\ of\ trials\ made)}$

## Experimental Probability Examples

Example 1:

Find the probability of getting a five when five has occurred 20 times when a dice is thrown 100 times.

Solution:

Let M be the event of getting a five.

Here total number of trials made = 100

And the number of times the outcome occurs = 20

Then $P (M)$ = $\frac{number\ of\ times\ of\ occurrence\ of\ the\ event}{number\ of\ trials}$

= $\frac{20}{100}$

= $\frac{1}{5}$

= 0.2

So the probability of getting a five in the experiment is 0.20.
Example 2:

The following chart that gives the marks of a total of 1200 students in an examination.

 Marks No. of Students 100-120 63 120-140 140 140-160 500 160-180 320 180-200 175

If we select a student then find the probability that the student has marks:

i) Above 180

ii) Between 140 and 180

iii) Under 120

Solution:

Clearly the total number of students is the total number of trials here which is equal to 1200.

i) From the chart we can see that there are 175 students that have marks above 180.

Hence P (marks above 180) = $\frac{175}{1200}$ = 0.15 (approximately)

ii) From the chart we can see the number of students getting marks between 140 and 180 are 500 + 320 = 820.

Hence P (marks between 140 and 180) = $\frac{820}{1200}$ = 0.68

iii) From the given chart we can see that number of students getting marks below 120 are 63.

Hence, P (marks under 120) = $\frac{63}{1200}$ = 0.05