 # Theoretical Probability

Probability is related to the calculation of the occurrence of a particular event. The value of probability is always expressed between 0 and 1.

For example: what is the likelihood that a two-sided one-rupee coin when tossed in the air? There are two possible outcomes, head, and tail.

### Theoretical Probability Definition

The theoretical probability math definition states that it is related to the theory behind probability. In theoretical probability, we utilise the knowledge of a situation to calculate the probability of an event. We do not conduct any experiment; instead, we just use the knowledge of a situation. The theoretical probability formula is as follows: it states that the probability of occurrence of an event is equal to the number of favourable outcomes divided by the total number of outcomes which are possible.

The mathematical formula of how we define theoretical probability is:

P(E)= $\frac{\text{The count of favourable outcomes}}{\text{Total number of possible outcomes}}$.

### Theoretical Probability Examples

Let us have a look at some theoretical probability questions:

Question 1) Find the probability that when a fair die is rolled, it rolls a 4

Answer) Here, the total number of possible outcomes is 6.

Number of favourable outcomes = number of times a fair die can roll to a 4 in a single throw = 1

According to the formula of theoretical probability, ‘

P(E)= $\frac{\text{The count of favourable outcomes}}{\text{Total number of possible outcomes}}$.

So, P (a fair die rolls a 4 in a throw) = 1/6

Question 2) A fair die is rolled. Find out the probability that the die rolls up to an odd number.

Answer) Here, the total number of possible outcomes is 6.

Number of favourable outcomes = number of times a fair die can roll to an odd number in a single throw.

Total outcomes of a fair die = {1,2,3,4,5,6}

Favourable outcomes (Odd numbers) = {1,3,5} =3

So, number of favourable outcomes = 3.

According to the formula of theoretical probability, ‘

P(E)= $\frac{\text{The count of favourable outcomes}}{\text{Total number of possible outcomes}}$.

So, P (die rolls up to an odd number) = 3/6 = 1/2

### What is Experimental Probability?

It is also known as empirical probability. It is calculated on the basis of performance of actual experiments or trials and its outcomes. Experiments are conducted in a serial manner. These are called random experiments as the results of these experiments are unpredictable. The experiments are carried out a number of times to determine the outcomes.

The mathematical formula of how we define experimental probability is:

P(E) = $\frac{\text{the number of times event E occurs}}{\text{ total number of trials of the experiment}}$.

### Solved Problems

Let us have a look at some experimental probability questions: -

Question) Two friends A and B toss a fair coin 10 times in a row. The outcomes for this experiment are as follows:

 Coin tossed by: Number of heads Number of tails A 5 5 B 2 8

Find the experimental probability for each outcome.

Answer) According to the formula of experimental probability,

P(E) = $\frac{\text{the number of times event E occurs}}{\text{total number of trials of the experiment}}$.

Now,

P (Occurrence of heads) = $\frac{\text{number of times head occurs}}{\text{total number of trials}}$

P (Occurrence of tails) = $\frac{\text{number of times tails occurs}}{\text{total number of trials}}$

## Calculation of Experimental Probability:

 Coin tossed by: Number of heads Number of tails Experimental probability for heads Experimental probability for tails A 5 5 5/10= 0.5 5/10 = 0.5 B 2 8 2/10 = 0.2 8/10 = 0.8

### Theoretical Probability vs Experimental Probability

when comparing experimental and theoretical probability, we should clearly look at their definitions to understand the fundamental difference between the two.

In the case of experimental probability, we perform experiments repeatedly to get to know the outcomes and calculate the probability of those series of outcomes. These experiments are known as random experiments as the results of these experiments are unpredictable. The collection of outcomes is what constitutes an event. If the outcomes have equal chances of happening, the event is termed as an equally likely event. Each repetition for conducting the experiment is called a trial. By the definition of probability, we can write this formula for the calculation of the probability of an event: P(E)= $\frac{\text{The count of favourable outcomes}}{\text{Total number of possible outcomes}}$

When in the case of experimental probability, the number of trials is extremely high, the experimental probability then starts approaching the theoretical probability values. The theoretical probability meaning is when the probability is calculated by utilising the knowledge of a certain situation and not carrying out the experiment actually.

In real life, there are some situations when carrying out experiments is not feasible, or it is too expensive to carry out those experiments. In such cases, theoretical probabilities are calculated to have a fair idea of how much likely are the outcomes to occur and so that necessary steps or precautions can be taken to avoid dangerous situations. For example, when we launch a satellite, the probabilities calculated are theoretical and not experimental.

Q1) What is the Sample Space of Outcomes When Two Coins are Tossed?

Answer) when any fair coin is tossed, the occurrence of the head or a tail is equally likely. When two coins are tossed the sample space of all possible outcomes is:

S = {HH.HT.TH.TT} (image will be uploaded soon)

Q2) What is a Zero Probability?

Answer) The probability of occurrence of any event lies between 0 and 1.

0 ≤ P(E) ≤ 1. When the chances of occurrence of an event are impossible, it can have a zero probability. For example, the sun rising from the west has a zero probability.

Contrary to that, when an event is certain to happen, the probability of that event is 1.

Q3) What is the Rule of Complements in Probability?

Answer) If P(E) is the probability that the event will occur and if P(E’) is the probability that the event will not occur, then

P(E) + P(E’) = 1

Here, P(E’) is the complement of P(E).