Conditional Probability Explained with Formula and Real Examples

Probability is a branch of Mathematics which deals with the study of occurrence of an event. There are several approaches to understand the concept of probability which include empirical, classical and theoretical approaches. The conditional probability of an event is when the probability of one event depends on the probability of occurrence of the other event. When two events are mutually dependent or when an event is dependent on another independent event, the concept of conditional probability comes into existence. 

Conditional Probability Definition:

Conditional probability of occurrence of two events A and B is defined as the probability of occurrence of event ‘A’ when event B has already occurred and event B is in relation with event A.

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The above picture gives a clear understanding of conditional probability. In this picture, ‘S’ is the sample space. The circles A and B are events A and B respectively. The sample space S is restricted to the region enclosed by B when event B has already occurred. So, the probability of occurrence of event A lies within the region of B. This probability of occurrence of event A when event be has already existed lies within the region common to both the circles A and B. So, it can be denoted as the region of A ∩ B.

Conditional Probability Examples:

• The man travelling in a bus reaches his destination on time if there is no traffic. The probability of the man reaching on time depends on the traffic jam. Hence, it is a conditional probability.

• Pawan goes to a cafeteria. He would prefer to order tea. However, he would be fine with a cup of coffee if the tea is not being served. So, the probability that he would order a cup of coffee depends on whether tea is available in the cafeteria or not. So, it is a conditional probability.

• It will rain at the end of the hottest day. Here, the probability of occurrence of rainfall is depending on the temperature throughout the day. So, it is a conditional probability. 

• In a practical record book, the diagrams are written with a pencil and the explanation is written in black ink. Here, the theory part is written in black ink irrespective of whether the diagrams are drawn with a pencil or not. So, the two events are independent and hence the probabilities of occurrence of these two events are unconditional.

Conditional Probability Formula:

The formula for conditional probability is given as:

P(A/B) = \[\frac{N(A\cap B)}{N(B)}\]

In the above equation,

P (A | B) represents the probability of occurrence of event A when event B has already occurred

N (A ∩ B) is the number of favorable outcomes of the event common to both A and B

N (B) is the number of favorable outcomes of event B alone.

If ‘N’ is the total number of outcomes of both the events in a sample space S, then the probability of event B is given as:

P(B) = \[\frac{N(B)}{N}\] → (1) 

Similarly, the probability of occurrence of event A and B simultaneously is given as:

P(A ∩ B) = \[\frac{N(A\cap B)}{N}\]→ (2)

Now, in the formula for conditional probability, if both numerator and denominator are divided by ‘N’, we get

P(A/B) = \[\frac{\frac{N(A\cap B)}{N}}{\frac{N(B)}{N}}\]

Substituting equations (1) and (2) in the above equation, we get 

P(A/B) = \[\frac{P(A\cap B)}{P(B)}\]

Conditional Property Problems:

Question 1) When a fair die is rolled, find the probability of getting an odd number. Also find the probability of getting an odd number given that the number is less than or equal to 4.

Solution:

In the given questions there are two events. Let A and B represent the 2 events.

A = Getting an odd number when a fair die is rolled

B= Getting a number less than 4 when a fair die is rolled

The possible outcomes when a die is rolled are {1, 2, 3, 4, 5, 6}

The total number of possible outcomes in this event of rolling a die: N = 6

For the event A, the number of favorable outcomes: N (A) = 3

For the event B, the number of favorable outcomes: N (B) = 4

The number of outcomes common for both the events: N (A ∩ B) = 2

The probability of event A is given as:

P(A) = \[\frac{N(A)}{N} = \frac{3}{6}\] = 0.5

The probability of occurrence of event A given event B is 

P(A/B) = \[\frac{N(A\cap B)}{N(B)} = \frac{2}{4}\] = 0.5.

Fun Facts:

• The conditional probability of two events A and B when B has already occurred is represented as P (A | B) and is read as “the probability of A given B”.

• The probability of occurrence of an event when the other event has already occurred is always greater than or equal to zero. 

• If the probability of occurrence of an event when the other event has already occurred is equal to 1, then both the events are identical.