Question

Suppose we have two weighted coins, one of which comes up heads with probability 0.1, and the other of which comes up heads with probability 0.7. Unfortunately, the coins are otherwise identical, and we have lost track of which is which. Suppose we flip a randomly chosen coin 11 times and let N be the random variable giving the number of heads seen. If in the first 4 flips we see 3 heads, what is the conditional expected number of heads in the 11flips ? E[N∣3 of first 4 are H]=

Answer 1

To find the conditional expected number of heads in the 11 flips given that 3 of the first 4 flips are heads, we need to consider the probabilities of the two coins. Using the law of total probability and the concept of conditional expectation, we can calculate the conditional expected number of heads as 7.669.

Step-by-step explanation:

To find the conditional expected number of heads in the 11 flips, given that 3 of the first 4 flips are heads, we need to consider the probabilities of the two coins. Let C1 be the coin that comes up heads with probability 0.1, and C2 be the coin that comes up heads with probability 0.7. We want to find E[N∣3 of first 4 are H], where N is the random variable representing the number of heads seen in the 11 flips.

The conditional expected number of heads can be calculated using the law of total probability and the concept of conditional expectation. Let A be the event that C1 is chosen, and B be the event that C2 is chosen. We have P(A) = P(B) = 0.5, since we have lost track of which coin is which. The probability of getting 3 heads in the first 4 flips given that C1 is chosen is (0.1)^3 * (0.9)^1 = 0.0009, and the probability of getting 3 heads in the first 4 flips given that C2 is chosen is (0.7)^3 * (0.3)^1 = 0.0735. Therefore, the conditional probability of C1 given that 3 heads are observed is P(A∣3 of first 4 are H) = P(A) * P(3 of first 4 are H∣A) / P(3 of first 4 are H) = 0.5 * 0.0009 / (0.5 * 0.0009 + 0.5 * 0.0735) = 0.005681818.

Similarly, the conditional probability of C2 given that 3 heads are observed is P(B∣3 of first 4 are H) = 0.5 * 0.0735 / (0.5 * 0.0009 + 0.5 * 0.0735) = 0.994318182.

Now, the conditional expected number of heads in the 11 flips can be calculated as E[N∣3 of first 4 are H] = P(A∣3 of first 4 are H) * E[N∣A] + P(B∣3 of first 4 are H) * E[N∣B].

For C1, the expected number of heads in the 11 flips is E[N∣A] = (0.1) * 11 = 1.1.

For C2, the expected number of heads in the 11 flips is E[N∣B] = (0.7) * 11 = 7.7.

Therefore, the conditional expected number of heads in the 11 flips given that 3 of the first 4 flips are heads is E[N∣3 of first 4 are H] = 0.005681818 * 1.1 + 0.994318182 * 7.7 = 7.669.