Biologists would like to catch Costa Rican glass frogs for breeding. There is 70% probability that a glass frog they catch is male. If 10 glass frogs of a certain species are caught, what are the chances that they will have at least 2 male and 2 female frogs? What is the expected value of the number of female frogs caught?

The "and two female frogs" is throwing me off as if it was just two male frogs I could just use $$P(X=2) = {10\choose 2}(0.7)^2(0.3)^8$$ but unsure how I would set it up for two male frogs and female frogs

  • $\begingroup$ For the first part, just enumerate the bad cases. There aren't many! Then compute the probability of each and subtract their sum from $1$. Expectation can be read off from the assumptions. $\endgroup$ – lulu Sep 14 '17 at 1:36

If you have ten frogs, then "at least two male and (at least) two female frogs", means that the count for male frogs is between $2$ and $8$.

Let the count be $X$. then you need to evaluate: $\mathsf P(2\leq X\leq 8)$ when $X\sim\mathcal {Bin}(10, 0.10)$

Here I suggest using the law of complementary probability.

$$\mathsf P(2\leq X\leq 8) ~=~1-\mathsf P(X\in \{0,1,9,10\})$$


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