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We are given that $P(A) = P(B) = 1/2$ and $P(A \cup B)= 2/3$.

I found that events $A$ and $B$ are not mutually exclusive by showing that $P(A \cup B) \neq P(A) + P(B)$ (i.e. $2/3 \neq 1$)

I also found that the two events are not independent by showing that $P(A \cap B) \neq P(A)P(B)$ (i.e. $1/3 \neq 1/4$)

Because the events are not mutually exclusive or independent I am having trouble solving for $P(A'\cap B)$ and $P(A' \cap B')$. I wanted to use the fact that $P(A'\cap B) = P(B) - P(A\cap B)$ but the events must be mutually exclusive to use this.

Any suggestions?

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Venn Diagrams are your friend. Draw one out, and the following should be easy to see:

First: $$ P(A\cup B)=P(A)+P(B)-P(A\cap B) $$ This tells you that $P(A\cap B)=P(A)+P(B)-P(A\cup B)$; you know all the quantities on the right side.

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

Third: $$ P(A^C\cap B^C)=1-P(A\cup B). $$

  • 0
    can I still use these equations even though events a and b are not mutually exclusive? – 2017-02-07
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    I mean, equations involving the intersection would be pretty boring if they only applied to mutually exclusive events haha – 2017-02-07