Question

In an urn containing 1000 numbered balls, each labeled from 1 to 1000, the following situations arise:

(a) If we draw 1000 balls with replacement, what is the approximate probability of observing the ball numbered 1 less than 3 times?

(b) When making 1000 picks with replacement, what is the expected number of times the ball numbered 1 is observed?

(c) If we draw 1000 balls without replacement, what is the probability that the ball numbered 2 is picked before the one numbered 51?

(d) When drawing only 50 balls out of the 1000 without replacement, what is the expected number of times the ball numbered 1 is observed?

Answer 1

The probability of observing the ball numbered 1 less than 3 times when drawing 1000 balls with replacement can be calculated using the binomial probability formula. The expected number of times the ball numbered 1 is observed when making 1000 picks with replacement can be calculated using the formula E(X) = np. The probability of picking the ball numbered 2 before the one numbered 51 when drawing 1000 balls without replacement can be calculated using the formula P(2 before 51) = P(2 on the first draw) + P(2 on the second draw). The expected number of times the ball numbered 1 is observed when drawing only 50 balls out of the 1000 without replacement can be calculated using the formula E(X) = np.

Step-by-step explanation:

(a)

To find the approximate probability of observing the ball numbered 1 less than 3 times when drawing 1000 balls with replacement, we can use the binomial probability formula. The probability of observing the ball numbered 1 less than 3 times can be calculated as P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2), where X follows a binomial distribution with n = 1000 and p = 1/1000. Substitute the values into the formula and calculate the probability.

(b)

The expected number of times the ball numbered 1 is observed when making 1000 picks with replacement can be calculated using the formula E(X) = np, where X follows a binomial distribution with n = 1000 and p = 1/1000. Substitute the values into the formula and calculate the expected number of times.

(c)

To find the probability that the ball numbered 2 is picked before the one numbered 51 when drawing 1000 balls without replacement, we can calculate the probability using the formula P(2 before 51) = P(2 on the first draw) + P(2 on the second draw). In order to calculate P(2 on the second draw), we need to subtract the probability of picking 51 before 2 from 1. Calculate the probabilities and subtract to get the probability of picking 2 before 51.

(d)

The expected number of times the ball numbered 1 is observed when drawing only 50 balls out of the 1000 without replacement can be calculated using the formula E(X) = np, where X follows a hypergeometric distribution with N = 1000, n = 50, and K = 1. Substitute the values into the formula and calculate the expected number of times.