Question

A balanced coin is launched 5 times. Let N be the number of consecutive heads, what is the standard deviation of N?

Answer 1

To find the standard deviation of the number of consecutive heads when a balanced coin is flipped 5 times, we need to determine all possible outcomes and their probabilities. The mean of N is 1.25 and the standard deviation is approximately 1.58.

Step-by-step explanation:

To find the standard deviation of the number of consecutive heads when a balanced coin is flipped 5 times, we need to determine all possible outcomes and their probabilities.

Let's define N as the number of consecutive heads.

Possible outcomes are:

1. 0 consecutive heads (HHHHH)
2. 1 consecutive head (THHHH, HTHHH, HHTHH, HHHTH, HHHHT)
3. 2 consecutive heads (TTHHH, HTTHH, HHTTH, HHHTT)
4. 3 consecutive heads (TTTHH, HTTTH, HHTTT)
5. 4 consecutive heads (TTTTH, HTTTT)
6. 5 consecutive heads (TTTTT)

Assuming the coin is fair, each outcome has an equal probability of occurring, which is 1/32.

Next, we'll calculate the mean (expected value) of N using the formula: E(N) = ∑(x * P(x)), where x is the value of N and P(x) is the probability of that value.

E(N) = (0 * 1/32) + (1 * 5/32) + (2 * 4/32) + (3 * 3/32) + (4 * 2/32) + (5 * 1/32) = 40/32 = 1.25

Finally, we'll calculate the standard deviation of N using the formula: sqrt(∑((x - E(N))^2 * P(x))).

Variance = (0.25 * 1/32) + (0.25 * 5/32) + (0.25 * 4/32) + (1.25 * 3/32) + (2.25 * 2/32) + (3.25 * 1/32) = 80/32 = 2.5

Standard deviation = sqrt(2.5) ≈ 1.58