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Suppose that A & b are mutually exclusive events. Then P(A)=.3 and P(B)=.5. What is the probability that either A or B occurs? A occurs but b doesn't. Both A and B occur.

1) Since the are mutually exclusive:

$P(A \cup B)= P(A)+P(B)=.3+.5=.8$

2) $A$ occurs $but$ B does not: $.3$

3) Both $A$ and $B$ occur:

Since they are mutually exclusive:

$P(A \cap B)= 0$ or the empty set

Are these correct

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  • $\begingroup$ I assume your question is to check your answers ... You may want to re-evaluate number 2 $\endgroup$
    – Bram28
    Feb 6, 2017 at 3:45
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    $\begingroup$ Looks good now! $\endgroup$
    – Bram28
    Feb 6, 2017 at 3:48
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    $\begingroup$ Note, "$P(A\cap B)=0$ or the emptyset" What is true is that $Pr(A\cap B)=0$ and that $A\cap B=\emptyset$. It is not true that $Pr(A\cap B)$ is the emptyset just as it is not true that $A\cap B=0$. The one is a set, the other is a number. $\endgroup$
    – JMoravitz
    Feb 6, 2017 at 5:27

2 Answers 2

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One way to think about 2)

We always have:

$P(A) = P(A\cap B) + P(A \cap B^C)$

but now that $A$ and $B$ are mutually exclusive, we have $P(A\cap B)=0$

thus: $P(A\cap B^C)=P(A)$

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I encountered this exact same problem on one of my homework assignments from the Introduction to Probability textbook by Charles M. Grinstead. To guide you in the right direction for part $2$), consider Corollary $2.1$ from his textbook: For any two events $A$ and $B$ (mutually exclusive in your case), we have $$P(A) = P(A \cap B) + P(A\cap B^c)$$ If you want to read more, turn to page $32$ on the following link for the book. https://math.dartmouth.edu/~prob/prob/prob.pdf

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