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 hunter independently hits his target with probability p, compute the expected number of ducks that escape unhurt when a flock of size 10 flies overhead.

Question

Mathematics

Goldie

5 August, 11:29

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 hunter independently hits his target with probability p, compute the expected number of ducks that escape unhurt when a flock of size 10 flies overhead.

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Gael Parrish · Answer 1

The expected number of ducks that escape unhurt is:

5

Step-by-step explanation:

First the process is binomial process. Having said this, we have 10 hunters, and 10 ducks flying overhead while the hunter take their shot at the same time.

By expected value, we mean average or mean. And the mean of a binomial distribution is:

Mean = > E (x) = np.

We must also know that there is only two probability possibilities:

1. The hunter hits the duck (p)

2. The hunter misses the duck (1-p) = q.

Hence;

p = 1/2, and (1-p) = 1/2.

Therefore;

E (x) = np; where n is the total number of hunter.

E (x) = 10 * (1/2) = 5

The expected number of ducks that escape unhurt is 5