Question

The owner of 5 retrievers throws a ball into a field, and one of the dogs brings it back. He repeats this action until every dog has retrieved the ball at least once. Assuming that each dog has the same chance of getting the ball with each throw, calculate the mean and variance of the number of throws required.

Answer 1

Answer 2

Let's calculate the mean and variance of the number of throws required for each dog to retrieve the ball at least once.

Let X be the number of throws required for a single dog to retrieve the ball at least once.

For a single dog to retrieve the ball at least once, we need to consider the following possibilities:

1. The first throw goes to a dog that has not retrieved the ball yet (probability: 5/5).

2. The second throw goes to a different dog that has not retrieved the ball yet (probability: 4/5).

3. The third throw goes to a different dog that has not retrieved the ball yet (probability: 3/5).

4. The fourth throw goes to a different dog that has not retrieved the ball yet (probability: 2/5).

5. The fifth throw goes to the remaining dog that has not retrieved the ball yet (probability: 1/5).

The number of throws required for a single dog to retrieve the ball at least once follows a geometric distribution with a probability of success p = 1/5 (since there are 5 dogs, and each has an equal chance of getting the ball). The mean (expected value) and variance of a geometric distribution are given by:

Mean (μ) = 1 / p Variance (σ^2) = (1 - p) / (p^2)

Let's calculate the mean and variance:

Mean (μ) = 1 / (1/5) = 5 throws Variance (σ^2) = (1 - 1/5) / ((1/5)^2) = 20 throws^2

Now, since each dog retrieves the ball independently, we can use the properties of expectation and variance for independent random variables to find the mean and variance for all 5 dogs.

Let Y be the total number of throws required for all 5 dogs to retrieve the ball at least once.

Since the number of throws for each dog follows the same geometric distribution with the same mean and variance, we have:

Mean of Y (μ_Y) = 5 * Mean (μ) = 5 * 5 = 25 throws Variance of Y (σ^2_Y) = 5 * Variance (σ^2) = 5 * 20 = 100 throws^2

So, the mean number of throws required for all 5 dogs to retrieve the ball at least once is 25 throws, and the variance is 100 throws^2.