The problem about the distribution of a random variable

[vilgeforc5]

A target with a probability of hitting p is fired until there are k hits. Find the distribution function and the mathematical expectation of the number of shots
it seems to me that it should be something like $$P=C_{n-1}^{k-1}*p^{k}*(1-p)^{n-k}$$ but I'm not sure. can anyone help?

 

[Steven G]

I could be wrong, but don't you need to use the definition of expected value to find the expected value?