1420 Hunters A and B shoot at a target with probabilities of

1.4-20. Hunters A and B shoot at a target with probabilities of p1 and p2, respectively. Assuming independence, can p1 and p2 be selected so that P(zero hits) = P(one hit)=P(two hits)?

Solution

We want to see if each of the hunters take a shot once, P(zero hits) = P(one hit)=P(two hits) is possible or not.

It is not possible. We will show this by contradiction.

Suppose it was possible.

Since there is either zero or one or two hits, then P(0 hits) = P(1 hit) = P(2 hits) = 1/3 .

Then,

0 = P(2 hits) P(0 hits) = p1p2 (1 p1)(1 p2) = p1 + p2 1.

Therefore p2 = 1 p1.

In order for p1*p2 = 1/3 ,

p1(1 p1) = 1/3

Now, the maximum value of the above function is 1/4 which is achieved at p1 = 1/2

Thus, there can\'t exist any value of p1 which satisfies the above expression.

Hence, it is not possible.

Hope this helps.