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Conditional Probability

  • If occurrence of one event (B) is influenced by the occurrence of another event (A), then the two events A and B are known as dependent events
    Let A and B be two events. Then, conditional probability of ‘B given A’ is the probability of happening of B when it is known that A has already happened. On the other hand, the probability of happening of B when nothing is known about the happening of A is called unconditional probability of B.
    The conditional probability of ‘B given A’ is denoted by P(B/A). The unconditional probability is P(B).
    Let P(A) > 0. Then, conditional probability of event ‘B given A’ is defined as
  • Description: 85572.png P(A) ≠ 0 and P(A) > 0
    If P(A) = 0, the conditional probability P(B/A) is not defined.
  • Description: 85566.png P(B) ≠ 0 and P(B) > 0

Two fair dice are rolled. If the sum of the numbers obtained is 4, find the probability that the numbers obtained on both the dice are even.
Let A be the event that the sum of the numbers is 4.
Let B be the event that the numbers on both the dice are even.
Here, we have to find P(B/A).
When two dice are rolled together, there can be 6 × 6 = 36 outcomes.
Event A has 3 favourable outcomes, namely (1, 3), (2, 2) and (3, 1)
Description: 85560.png 
Event (A ∩ B), i.e., the event that the sum of the numbers is 4 and the numbers are even, has 1 favourable outcome, namely (2, 2).
∴ P [sum 4 and numbers even] Description: 85554.png
We know that Description: 85548.png 
Thus, P [Numbers even given sum is 4]Description: 85542.png
Description: 85536.png 


Note: If A and B are two independent events, then
P(A ∩ B) = P(A) × P(B) or P(B/A) = P(B) or P(A/B) = P(A)

Multiplication Rule or Compound Probability or Joint probability

The probability of occurrence of two events A and B simultaneously is known as the compound probability or joint probability of A and B. Denoted by P(A  B).


Let A and B be two events with respective probability P(A) and P(B). Let P(B/A) be the conditional probability of event B, given that event A has happened. Then, the probability of simultaneous occurrence of A and B is


P(A  B) = P(B) · P(A/B) or P(A  B) = P(A) · P(B/A) (when the events occur together)

A box has 1 red and 3 white ball. Two balls are drawn one after the other from the box. Find the probability that the two balls drawn would be red if the first ball drawn is returned to the box before the second ball is drawn.
Let event A: the first ball drawn is red
Event B: the second ball drawn is red
Here, Description: 85530.png Also, since the first ball is returned before the second ball is drawn,
Description: 85524.png 
∴ P [Two balls are red]Description: 85518.png
Description: 85512.png 
Description: 85506.png 


Note: De Morgan’s law for probability:
P(A′ ∩ B′) = P(A ∪ B)′ = 1 − P(A ∪ B)
P(A′ ∪ B′) = P(A ∩ B)′ = 1 − P(A ∩ B)
Probability that only event A occurs and event B does not occur is
Description: 89087.png

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