Question

A fair coin is tossed three times. The random variable X is defined to be 2h, where h is the number of flips that come up heads. For example, X(HHT) = 2². What is E(X)?

Answer 1

The expected value E(X), for a fair coin that is tossed three times where X equals twice the number of heads, is 3.25.

Step-by-step explanation:

To calculate the expected value E(X) for a fair coin tossed three times, we must first list the possible outcomes and compute the probabilities for each possible number of heads. The sample space for the random variable X is as follows:

• 0 heads (TTT): X = 0
• 1 head (THT, TTH, HTT): X = 21 = 2
• 2 heads (THH, HTH, HHT): X = 22 = 4
• 3 heads (HHH): X = 23 = 8

We calculate the expected value by multiplying each outcome by its probability and summing them up. Since the coin is fair, the probability for each outcome of heads (0, 1, 2, or 3) is determined based on the number of ways it can occur divided by the total number of possible outcomes (23 = 8 for three coin tosses).

The probabilities are:

• P(0 heads) = 1/8 (one way to get 0 heads)
• P(1 head) = 3/8 (three ways to get 1 head)
• P(2 heads) = 3/8 (three ways to get 2 heads)
• P(3 heads) = 1/8 (one way to get 3 heads)

Thus, the expected value E(X) is:

E(X) = (0 × 1/8) + (2 × 3/8) + (4 × 3/8) + (8 × 1/8)

E(X) = 0 + 0.75 + 1.5 + 1

E(X) = 3.25

Therefore, the expected value of X, when a fair coin is tossed three times, is 3.25.