Ceasar's Mind

Here’s an interesting problem that might be useful to know in the future.

Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time,, but each chooses his target at random, independently of the others. If each hunters independently hits his target with probability p, compute the expected number of ducks that escape unhurt when a flock of size 10 flies overhead.

Solution

Let X_i equal 1 if the i-th duck escapes unhurt, and 0 otherwise, for i = 1, 2, …, 10. The expected number of ducks to escape can be expressed as:

E[X_1 + ... + X_10] = E[X_1] + … + E[X_10]

To compute E[X_i] = P{X_i = 1], we note that each of the hunters will independently, hit the the i-th duck with probability p/10, so: P{X_i=1} = (1 – (p / 10)) ^ 10 (that is, the chance that the duck escapes unharmed is equal to the chance that none of the hunters hit it). Hence, E[X] = 10 * (1 – (p / 10)) ^ 10.

(Just for reference, if p = 1, E[X] = 3.49.)

More generally, E[X] = T * (1 – (p / T)) ^ S, where T represents the number of targets, and S the number of shooters.

This is pretty cool in my opinion, as the problem shows up a lot in various games (although a lot of games use algorithms to determine which unit to fire at rather than chance), Tower Defense games perhaps being the most obvious.