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There are $7$ people in a room. $4$ male & $3$ female. $2$ persons are chosen randomly. The probability of "doing something right" if $2$ females are chosen is $0.6\,$. When 2 males are chosen the probability of "doing something right" is $0.1$ and when a male and a female are chosen the probability of "doing something right"is $0.3\,$.

A) What's the Expected Value of the probability on successful attempts

B) What's the Expected Value of the probability on successful attempts if $1$ male is chosen certainly.

*forgive me if I'm not clear enough cause this is a translated problem.

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There are $\binom{7}{2}$ equally likely ways to choose $2$ people. There are $\binom{3}{2}$ ways to choose two females, and $\binom{3}{1}\binom{4}{1}$ ways to choose one female and one male, and $\binom{4}{2}$ ways to choose two males.

Thus the (simplified) probability of two females is $\frac{1}{7}$, the probability of one of each is $\frac{4}{7}$, and the probability of two males is $\frac{2}{7}$.

A) The probability that things turn out well is therefore $$(0.6)(1/7)+(0.3)(4/7)+(0.1)(2/7).$$

B) Here I will be making an interpretation of the question. I assume that we are asked to find the probability that things turn out well, given that at least one male is chosen. Let $p$ be the probability that one of each is chosen, given that at least one male is chosen. By the usual conditional probability calculation, we have $p=\frac{4/7}{6/7}$, that is, $2/3$. So the probability two males are chosen, given that at least one is chosen, is $1/3$. Thus the probability of success, given that at least one male was chosen, is $$(0.3)(2/3)+(0.1)(1/3).$$

Remark: Note that the interpretation is quite sensitive to the exact wording, and therefore to possible small inaccuracies of translation.

For example, suppose that we are told that the people were chosen one at a time, and that the first person chosen is male. Then given this information, the probability that one is male and one is female is $3/6$, not $2/3$. The calculation will also be different if we are told that one specific male, say Adam, was chosen.

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