Question

An underground military installation is fortified to the extent that it can withstand up to three direct hits from air-to-surface missiles and still function (this means the installation is destroyed with the fourth direct hit). Suppose an enemy aircraft is armed with missiles, each having a 30% chance of scoring a direct hit.

a) What is the average number of missiles that have to be fired to score the first direct hit?

b) What is the average number of missiles that have to be fired to destroy the installation?

c) What is the probability the installation is still functional after ten missiles have been fired?

d) What is the probability that the installation will be destroyed with the seventh missile fired?

e) What is the probability that it takes more than five missiles to score the first direct hit?

f) Suppose five missiles have been fired at the installation and none scored a direct hit. What is the probability it will take more than ten missiles total to score the first direct hit?

g) Suppose that ten missiles have been fired at the installation. What’s the probability that there were no direct hits?

h) Calculate the Poisson approximation to the probability gotten in part g. This approximation is not very close to the true answer gotten in part g. Why is that?

Answer 1

the probability of scoring the direct hit: p=0.3

a)This can be solved by geometric distribution.

the average number of missiles that have to be fired to score the first direct hit =

b) installation can be destroyed by 4 hits

The probability of having 4 hits is =

the average number of missiles that have to be fired to destroy the installation is geometric average =

c) the probability the installation is still functional after ten missiles have been fired = the probability that there is no hit or one hit or two hit or three hit in 10 missiles fired

=

d) This means that the 4th hit is on the 7th missile fired.

the probability that the installation will be destroyed with the seventh missile fired =