Conditional Probability: Overview, Questions, Preparation

Probability 2021 ( Probability )

Rachit Kumar Saxena

Rachit Kumar SaxenaManager-Editorial

Updated on May 12, 2021 06:18 IST

What is Conditional Probability?

When the occurrence of one event is related to the occurrence of another event, you can find its probability with conditional probability. If event A occurs only if event B occurs first, it is called conditional probability. It is represented as P(A|B).

The Formula of Conditional Probability

Conditional probability P(A|B) can be expressed mathematically as:

P(A|B) = N(A ∩ B)/N(B),

where N(A ∩ B) represents the elements that are shared by both A and B and N(B) are the elements that are in B. N(B) can never be 0.

If N is the number of elements in the sample space, then the below statement is also true:

P(A|B) = N(A ∩ B)/N / N(B)/N ---- (1)

The probability of any event is the ratio obtained by dividing the number of expected outcomes with the total outcomes. 

Therefore, N(A ∩ B)/N = P (A ∩ B) and N(B)/N = P(B) ----- (2)

From statements 1 and 2, we get:

P(A|B) = P (A ∩ B)/P(B)

Properties of Conditional Probability

Suppose E and F be the events that belong to the sample space S. 

1st Property: 

P(S|F) = P (S ∩ F)/P(F) = P(F)/P(F) = 1

and P(F|F) = P (F ∩ F)/P(F) = P(F)/P(F) = 1

Therefore, P(S|F) and P(F|F) = 1.

2nd Property:

If the sample space comprises events A and B, then the below statement holds true:

P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F).

3rd Property: 

If M and M′ are mutually exclusive or disjoint events, then the following statement is true:

P(M′|F) = 1 - P(M|F)

Weightage of Conditional Probability

You will learn conditional probability in detail in Class XII in ‘Probability’. The chapter has a weightage of around 10 marks, which means that conditional probability will have a maximum weightage of around 4 to 5 marks. 

Illustrative Examples on Conditional Probability

1. Find the value of P(A|B) if P(A) = ⅚ and P(B) = 7/6 and P (A ∩ B) = ⅙ 

Solution. P(A|B) = P(A ∩ B)/P(B) = ⅙ / 7/6 = 1/7 

2. A dice is thrown twice, and the numbers appearing are observed. If the sum of the numbers appearing on the dice is 5, what is the probability of number 2 appearing at least once?

Solution. Suppose that F is the event when the sum of the numbers is 5 

F = { (1,4), (2,3), (3,2), (4,1)} 

Therefore, P(F) = 4/36 = 1/9

Now, let E be the event that the number 2 appears at least once. 

E = { (1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)} 

P(E) = 11/36

E ∩ F = {(2.3), (3.2)}

P(E ∩ F) = 2/36 = 1/18 

Therefore, 

P(E|F) = P(E ∩ F)/P(F) = 1/18 /  1/9 = ½.

FAQs on Conditional Probability

Q: What are disjoint events?

A: Disjoint events are independent events, i.e., one event’s occurrence is not dependent on another event’s occurrence.  

Q: Who found the concept of conditional probability?

A: Mathematician Thomas Bayes determined the concept of conditional probability. 

Q: What is the difference between joint probability and conditional probability?

A: Joint probability is when two events happen simultaneously, whereas conditional probability is when an event occurs only when another event has already occurred.   

Q: What is the Bayes Theorem?

A: Bayes Theorem allows us to find the probability of an event when it is related to some condition. It is also known as conditional probability as it is very closely related to it. 

Q: Give the formula for Bayes Theorem.

A: The formula for Bayes Theorem is P(A|B) = P(B|A) P(A)/P(B). 
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