Q:

Combined Binomial Distribution Problem.

I have the following problem:
70% of women respond positively to a test, while only 40% of men do so. If 10 participants are selected (5 women and 5 men), what is the probability that only 1 man responds negatively?
Now what I have done is to have two distributions, one for women and one for men. Then I calculated P(Y=5) for women with p=0.7 and P(Y=4) for men with p=0.4, and since the events are mutually exclusive I´ve added the two, having a result of ~24%. Can this be correct...?

A:

The events of $5$ women out of $5$ respond positively and $4$ men out of $5$ respond positively are not mutually exclusive.  In fact it is perhaps reasonable to assume that they are independent, and so you can multiply the probabilities.

