A and B throw a die 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?a)Rs. 70 and Rs. 40b)Rs. 60 and Rs. 50c)Rs. 75 and Rs. 35d)noneof theseCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Expectation is a measure of the average value or outcome of a random variable. In this case, we want to find the respective expectations of A and B in the game they are playing.

Let's analyze the game step by step to determine the expectations of A and B.

- A starts by throwing the die. The probability of getting a 1 is 1/6, and the probability of not getting a 1 is 5/6.
- If A does not get a 1, it is now B's turn to throw the die. Similarly, the probability of B getting a 1 is 1/6, and the probability of not getting a 1 is 5/6.

Now, let's calculate the respective expectations of A and B.

- A's Turn:
- If A gets a 1 on the first throw, A wins Rs. 110.
- If A does not get a 1 on the first throw, the game continues to B's turn.
- The expected value of A's turn can be calculated as follows:
- E(A) = (1/6) * 110 + (5/6) * E(B)
- The first term represents the probability of A winning Rs. 110 on the first throw.
- The second term represents the probability of not winning on the first throw, and the game continues to B's turn.

- B's Turn:
- If B gets a 1 on the first throw, B wins Rs. 110.
- If B does not get a 1 on the first throw, the game continues to A's turn.
- The expected value of B's turn can be calculated as follows:
- E(B) = (1/6) * 110 + (5/6) * E(A)
- The first term represents the probability of B winning Rs. 110 on the first throw.
- The second term represents the probability of not winning on the first throw, and the game continues to A's turn.

Now, we can substitute the value of E(B) into the equation for E(A) and solve the system of equations to find the respective expectations of A and B.

By solving the equations, we find that E(A) = Rs. 60 and E(B) = Rs. 50.

Therefore, the respective expectations of A and B are Rs. 60 and Rs. 50, which matches option B.

Answer

Answer

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