# Theoretical Probability

Probability is taken under practice in order to describe uncertain events associated to an experiment. An experiment simply means a process like flipping a coin. An event is that can happen in an experiment like occurrence of head or tail. The results head or tails are the possible outcomes. This implies that a single result of the experiment is an outcome of it.

Probability is defined as the likeliness of the occurrence of an event. Probability always lies between 0 and 1.
If all the outcomes of an experiment have equal chance of occurrence then we can easily evaluate the event’s theoretical probability in an exact manner.

The probability can then be calculated by finding out the ratio of the number of outcomes that are favorable to the event and the total number of outcomes of the experiment.

In theoretical probability we are relying on the nature of the experiment and the event and not the actual conducting of the experiment. So what we are getting from calculating a theoretical probability is simply an expectation as to what can happen and to what extent can it happen.

## Formula

Let the outcomes associated with an experiment be represented in a set $S$ called the sample space.

Then $n (S)$ gives the number of total outcomes.

Let the favorable outcomes of the event be represented in set A. Then n (A) given the number of favorable outcomes of the event.

Then before calculating the theoretical probability it is a must check that all outcomes of the experiment has an equal chance of occurrence. Only then we proceed to calculation. If the outcomes are not equally likely to occur then we do not calculate the theoretical probability.
When the chances are equally likely then the theoretical probability is given by:

$\frac{n (A)}{n (S)}$

## Examples

Let us see some examples for better understanding.

Example 1: An urn contains 6 green, 3 white, 2 blue and 1 red balls. If a ball is picked then find the probability that ball is:

i) Red in color
ii) Either blue or white in color
iii) Not at all green in color

Solution: Here n (S) = 12

All the outcomes are equally likely to occur. So we can calculate the theoretical probability.

i) Favorable outcomes for a red ball = 1.

Hence, P (red) = $\frac{1}{12}$

ii) Favorable outcome for blue ball = 2

Favorable outcomes for white ball = 3.

Hence P (blue or white) = P (blue) + P (white) = $\frac{2}{12}$ + $\frac{3}{12}$ = $\frac{5}{12}$

iii) Favorable outcomes for green ball = 6

So, P (green) = $\frac{6}{12}$ = $\frac{1}{2}$

Hence, P (not green) = 1 - P (green) = 1 - $\frac{1}{2}$ = $\frac{1}{2}$

Example 2: Consider the experiment of the score board for a soccer team.

Find the theoretical probability of getting a score of 2 - 0.

Solution: Here, we do not know the limit of the scoreboard outcomes possibilities so the sample space is infinitely large.

Also the chances for getting a high score like 10 - 11 are very less and 12 -15 is almost rare, this implies that all outcomes do not have an equally likely chance to occur. In such a case thus we cannot find our theoretical probability.

Example 3: A pair of dice is cast. Find the theoretical probability that the sum of the numbers facing up is 5.

Solution: Let S be the sample space, which is set of all pairs of numbers 1 through 6.
Total number of events in a sample space = n(S) = 36

Let A be the set of favorable outcomes = {(1,4), (2,3), (3,2), (4,1)}

=> n(A) = 4

Now P(A) = $\frac{n(A)}{n(S)}$ = $\frac{4}{36}$ = $\frac{1}{9}$.