A shooter shoots at three targets until all of them are hit. The probability of hitting a target with one shot is $p$.

a) Find the probability that exactly 5 shots will be required.

b) Find the expected number of shots.
a) The desired event occurs only if the first four shots result in two hits, and the last shot is successful. The probability of this is $C_{4}^{2} p^{2}(1-p)^{2} \cdot p=6 p^{3}(1-p)^{2}$.

b) Given the probability of a hit $p$, the average number of shots needed to hit the first target is $1 / p$. This is intuitively clear and strictly follows from the properties of the geometric distribution. Therefore, on average, another $1 / p$ shots will be needed for the second target and the same for the third.

## Answer

a) $6 p^{3}(1-p)^{2} ;$ b) $3 / p$.

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