Question

suppose that we put 10 balls randomly into 10 boxes. that is, for each ball, we randomly choose any one of the 10 boxes, independently of the previous selections. let the random variable x represent the number of empty boxes. what is e(x)

Answer 1

The expected number of empty boxes when placing 10 balls into 10 boxes at random can be calculated using the concept of expected value. By defining an indicator variable for each box and calculating its expected value, we determine that E(X) = 10 * (9/10)^10.

Step-by-step explanation:

The student's question relates to the concept of expected value in probability, specifically concerning the expected number of empty boxes when placing 10 balls into 10 boxes at random.

To find the expected number of empty boxes E(X), we can define an indicator random variable Ij for each box, such that Ij = 1 if the j-th box is empty and 0 otherwise. Consequently, the number of empty boxes X can be defined as the sum of Ij for j = 1 to 10.

To find the expected value of Ij, note that the probability a given box is empty after placing one ball is (9/10), since each ball has a 1/10 chance of being placed in any other box. Since the placement of each ball is independent, the probability that a box is empty after placing all 10 balls is (9/10)^10.

Thus, E(Ij) = (9/10)^10 for each j. As there are 10 boxes, the expected number of empty boxes E(X) is thus 10 * (9/10)^10.