Multiple choice

A and B throw a dice in succession to win a bet with A starting first. Whoever throws '1' first wins an amount of Rs. 110. What are the respective expectations of A and B?

  1. Rs. 70 and Rs. 40

  2. Rs. 60 and Rs. 50

  3. Rs. 75 and Rs. 35

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Prob(A wins) = 1/6 + (5/6)(5/6)(1/6) + ... = (1/6) / (1 - 25/36) = (1/6) / (11/36) = 6/11. Prob(B wins) = 1 - 6/11 = 5/11. Expectation A = 6/11 * 110 = 60. Expectation B = 5/11 * 110 = 50.

AI explanation

The probability of either player winning on a single throw is 1/6. A's expectation is the total amount multiplied by his overall probability of winning, where A wins if he throws a 1 on his first attempt, or if both players fail on their first attempts and A eventually wins. A's probability of winning satisfies P(A) = 1/6 + (5/6)^2 * P(A), which gives P(A) = 6/11. Multiplying A's probability by Rs. 110 yields an expectation of Rs. 60, and since the combined expectations must equal Rs. 110, B's expectation is Rs. 110 - Rs. 60 = Rs. 50.

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