When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let…
2006
When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that successive tosses are independent, the expected value of N is
- A.
1/p
- B.
1/(1−p)
- C.
1/p^2
- D.
1/(1-p^2)
Attempted by 2 students.
Show answer & explanation
Correct answer: A
Answer: 1/p
Reasoning: N is the number of tosses until the first Head. The probability that the first Head occurs on the k-th toss is (1 − p)^{k−1} p for k = 1, 2, 3, ....
Write the expectation as a series: E[N] = sum_{k=1}^∞ k (1 − p)^{k−1} p.
Let r = 1 − p. Use the known series identity sum_{k=1}^∞ k r^{k−1} = 1/(1 − r)^2 for |r| < 1.
Substitute r = 1 − p to get E[N] = p * 1/(1 − (1 − p))^2 = p * 1/p^2 = 1/p.
Thus the expected number of tosses until the first Head is 1/p.