# Tricky Conditional Expectation

A group of 21 women and 14 men have a certain disease with probability p, independently. If we know that exactly 5 of the people have the disease, what is the expected number of women who have the disease?

Using LOTE here, we can assign an indicator random variable for those with the disease and condition on the probability that they are a woman with the disease. Let I be this indicator and W$_d$ be woman with the disease. However, I'm confused as to how to apply the law of total expectation from this. Help?

• Please add the self-study tag as this looks like an exercise connected to a course or textbook. – Xi'an Nov 13 '17 at 17:07
• Convince yourself that the probability of any single individual having the disease is 5/(21+14). Now use linearity of expectation. – Alex R. Nov 14 '17 at 1:04