Suppose that 67% of the students at a college are female and 50% are white. Therefore, if F is the event that a randomly chosen student is female, and W is the event that a randomly chosen student is white, then Pr(F) = 0.67 and Pr(W) = 0.5. Assuming that F and W are not necessarily independent events, what is the maximum percentage of the student body that could be both white and female?

Question

anonymous · Answer

Since there is no information given that says event W can't be completely inside of event F, you could assume that there is a chance all of the white students are also female. Therefore, the maximum percentage P(F and W) = 0.50

mathmate · Answer

$P(F\cap W)=P(F)+P(W)-P(F\cup W)$
$P(F\cap W)=0.67+0.5-P(F\cup W)$
For $P(F\cap W)$ to be a maximum, $P(F\cup W)$ must be a minimum, or 0.67.
Therefore
$P(F\cap W)=0.67+0.5-0.67=0.5$   exactly as @derrowap had it.

anonymous · Answer

@mathmate  First line: Why does F and W = F+W-(F or W) -- doesn't that just leave F or W?

mathmate · Answer

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The original equation is
F or W = F + W - F and W   because F and W is counted twice in F+W.
We transpose it to get what I posted.
I hope that explains it.
See also the drawing above.