Question

there are 2 bears, one black one white.
What is the probability that both are male?
What is the probability that one is male?
Given that the black one is male, what is the probability that the other is male?

Answer 1

Do we assume a \(\dfrac{1}{2}\) probability for the particular sex of the bears?

Answer 2

yes

Answer 3

In that case, since we have two distinct outcomes (male vs female) we can assume that sex follows a binomial distribution with each event having the same probability of \(\dfrac{1}{2}\).

The probability that both bears are male would then be given by
\[\binom{\color{red}2}{\color{blue}2}\left(\frac{1}{2}\right)^{\color{blue}2}\left(1-\frac{1}{2}\right)^{\color{red}2-\color{blue}2}\]
because of the \(\color{red}2\) bears we consider the event of having \(\color{blue}2\) males.

The second probability is computed similarly, but this time we want \(\color{red}1\) male bear.

The third probability involves a conditional event. Recall that
\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]
For independent events \(A\) and \(B\), we have
\[P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{P(A)P(B)}{P(B)}=P(A)\]
Since fur color and sex are presumably independent events, the probability that the white bear is male is the same as the probability as any one bear being male. In other words,
\[P(\text{white is male }|\text{ black is male})=\frac{P(\text{both bears are male})}{P(\text{black is male})}=\frac{\text{2nd prob}}{1/2}\]