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- Apr 14, 2013

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I am looking the following concering boolean algebra.

For a certain test group, it is found that $42\%$ of the people have never skied yet, that $58\%$ of them have never flown yet, and that $29\%$ of them have already flown and skied.

Which probability is higher then:

To meet someone, who has already skied, from the group of those who have never flown, or to find someone who has already flown from the group of those who have already skied?

I have done the following:

Let $S$ be the event that someone has already skied and let $F$ be the event that someone has already flown.

Then it is given that $P(\overline{S})=42\%$, $P(\overline{F})=58\%$ and $P(S\land F)=29\%$, or not?

From these probabilities, we get also $P(S)=100\%-P(\overline{S})=100\%-42\%=58\%$ and $P(F)=100\%-P(\overline{F})=100\%-58\%=42\%$.

We are looking for the conditional probabilities $P(S\mid \overline{F})$ and $P(F\mid S)$, or have I understood that wrong?

We have that \begin{align*}&P(S\mid \overline{F})=\frac{P(S\land \overline{F})}{P(\overline{F})} \\ &P(F\mid S)=\frac{P(F\land S)}{P(S)}=\frac{29\%}{58\%}=50\%\end{align*}

How can we calculate $P(S\land \overline{F})$ ?