Question 1026104: The probability that Jose shopping for himself will buy a tie is 0.2, the probability that he will buy a shirt is 0.3, and the probability that he will buy a tie given that he buys a shirt is 0.4. Find the probability that he will buy
1. both a shirt and a tie.
2. a shirt, a tie, or both.
Answer by mathmate(429)   (Show Source):
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Question:
The probability that Jose shopping for himself will buy a tie is 0.2, the probability that he will buy a shirt is 0.3, and the probability that he will buy a tie given that he buys a shirt is 0.4. Find the probability that he will buy
1. both a shirt and a tie.
2. a shirt, a tie, or both.
Solution:
The solution of the problem hinges on the definition of conditional probability, which is
P(A|B)=P(A∩B)/P(B)
We're given
A=event that he buys a tie
B=event that he buys a shirt
P(A)=0.2
P(B)=0.3
P(A|B)=P(A∩B)/P(B)=0.4
and the relationship between the union and the intersection:
P(A∪B)=P(A)+P(B)-P(A∩B)
Hence, using the definition above,
1. Both a tie and a shirt (P(A∩B)
P(A|B)=0.4
=P(A∩B)/P(B)
P(A∩B)
=P(A|B)*P(B)
=(P(A|B)/P(B))*P(B)
=0.4*0.3
=0.12
2. A shirt, a tie or both, P(A∪B)
P(A∪B)
=P(A)+P(B)-P(A∩B)
=0.2+0.3-0.12
=0.38
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