Question

A coin is tossed until one head appears or a tail appears \(4\) times in succession. The probability distribution of the number of tosses is

• A. A
• B. B
• C. C
• D. D

Hint:

In such stopping-time problems, list all possible terminal sequences clearly first. That makes the probability distribution easy to build.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The random variable \(X\) is the number of tosses. The experiment stops at Head (\(H\)) or after \(4\) tails (\(T\)).
Step 2: Key Formula or Approach:
Probabilities of single toss: \(P(H) = 1/2, P(T) = 1/2\).
Step 3: Detailed Explanation:
\(P(X=1) = P(H) = 1/2\).
\(P(X=2) = P(TH) = 1/2 \times 1/2 = 1/4\).
\(P(X=3) = P(TTH) = (1/2)^3 = 1/8\).
\(P(X=4)\): Either \(TTTH\) (head at \(4^{\text{th}}\)) or \(TTTT\) (stops due to limit).
\(P(X=4) = (1/2)^4 + (1/2)^4 = 2/16 = 1/8\).
Step 4: Final Answer:
Matches option (D).