Question

One roll of a fair die produces one of the numbers, 1, 2, 3, 4, 5, 6, with equal probabilities of 1/6 each. Let Yn be the sum of n independent rolls of a fair die. Find the limit as n approaches infinity of the probability that Yn is a multiple of 4.

Answer 1

The probability that the sum of n independent rolls of a fair die is a multiple of 4 approaches 0 as n approaches infinity.

Step-by-step explanation:

The probability that the sum of n independent rolls of a fair die, denoted as Yn, is a multiple of 4 can be found by calculating the probability of each possible sum that is a multiple of 4 and taking the limit as n approaches infinity. Let's consider the possible sums:

If n is even, then the possible sums are multiples of 2 (2, 4, 6, ...). In this case, the only way for the sum to be a multiple of 4 is if all the rolls are even numbers. Since each roll has a 1/2 chance of being even, the probability of the sum being a multiple of 4 is (1/2)^n.

If n is odd, then the possible sums are odd multiples of 2 (3, 5, 7, ...). In this case, it is not possible for the sum to be a multiple of 4, so the probability is 0.

Therefore, the limit as n approaches infinity of the probability that Yn is a multiple of 4 is 0.