Question

A fair coin is tossed 2 times. A person receives \( X^3 \) if he gets \( X \) number of heads. His expected gain is

• A. ₹ 2.00
• B. ₹ 1.00
• C. ₹ 2.50
• D. ₹ 5.20

Hint:

For expected value problems, use the formula \( E(X) = \sum P(X) \times X \), and apply the powers if necessary for the random variable.

Answer 1

The Correct Option is C

Step 1: Calculate the probabilities of each outcome.
There are 4 possible outcomes for tossing a coin twice: - 0 heads: Probability \( P(0) = \frac{1}{4} \) - 1 head: Probability \( P(1) = \frac{1}{2} \) - 2 heads: Probability \( P(2) = \frac{1}{4} \) Step 2: Find the expected gain.
The expected gain is given by: \[ E(X) = \sum_{X} P(X) \times X^3 \] Substituting the values, we get: \[ E(X) = \frac{1}{4}(0^3) + \frac{1}{2}(1^3) + \frac{1}{4}(2^3) = 0 + \frac{1}{2} + \frac{8}{4} = 2.50 \] Step 3: Conclusion.
The correct answer is (C) ₹ 2.50.