The problem I am currently working on is:
Consider the following information: where
- $A$={Visa Card}
- $B$={MasterCard}
$P(A)=0.5$, $P(B)=0.4$, and $P(A \cap B) = 0.25$
The part I am having difficulty with is part (e):
Given that an individual is selected at random and that he or she has at least one card, what is the probability that he or she has a Visa Card?
For some reason it is just eluding me. Could someone help me?
Best Answer
The probability of having at least one card is: $$P(A)+P(B)-P(A\cap B) = 0.65 $$
Denote $C$={ at least one card }.
The probability you need (definition of conditional probability):
$$ P(A\;|\;C) = \frac {P(A\cap C)}{P(C)}$$
If you have a Visa card, you have at least one card, so $P(A\cap C)=P(A)$
$$ P(A\;|\;C) = \frac {P(A)}{P(C)} = \frac{0.5}{0.65}$$