Question

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.

Answer 1

The expected number of ducks escaping unhurt is calculated by raising the probability of a single duck not being hit by one hunter, (1 - p), to the power of the number of hunters, which is 10, and then multiplying this by the total number of ducks, which is also 10.

Step-by-step explanation:

The expected number of ducks that escape unhurt when a flock of size 10 flies overhead, with each of ten hunters independently hitting their target with probability p, can be found by considering the probability of each duck escaping. Since each duck is targeted independently by each hunter, the probability that a specific duck is not hit by a particular hunter is 1 - p. The probability that a specific duck is not hit by any of the hunters is (1 - p)^{10}, because we must consider the scenario where all hunters miss this duck.

The expected number of ducks that escape is the sum of the probabilities that each individual duck escapes, which is 10 * (1 - p)^{10}. This is because there are 10 ducks, and each has the same probability of escaping unhurt.

Answer 2

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