Question

Bob, Sally and Linda roll a pair of dice in turn, always with Bob rolling first, Sally rolling next and then Linda's turn. Bob's objective is to obtain a sum of 5, Sally's objective is to obtain a sum of 6 , and Linda's objective is to obtain a sum of 8 . The game ends when one player reaches his or her objective, and that player is declared as the winner. Compute the expected number of turns until the game is over.

Answer 1

The expected number of turns until the game is over is 23.4 turns.

Step-by-step explanation:

To find the expected number of turns until the game is over, we need to calculate the probability of each player achieving their objective on each turn. Let's start with Bob. There are 4 ways to get a sum of 5 when rolling two dice: (1,4), (2,3), (3,2), and (4,1). Since there are 36 possible outcomes when rolling two dice, the probability of Bob getting a sum of 5 on a given turn is 4/36 = 1/9.

Similarly, Sally has 5 ways to get a sum of 6: (1,5), (2,4), (3,3), (4,2), and (5,1). So, the probability of Sally getting a sum of 6 on a given turn is 5/36 = 5/36.

Linda has 5 ways to get a sum of 8: (2,6), (3,5), (4,4), (5,3), and (6,2). Therefore, the probability of Linda getting a sum of 8 on a given turn is 5/36 = 5/36.

To find the expected number of turns for each player to achieve their objective, we divide the desired outcome by the probability of achieving it. So, for Bob, the expected number of turns is 1/(1/9) = 9. Similarly, for Sally, the expected number of turns is 1/(5/36) ≈ 7.2, and for Linda, it is also 1/(5/36) ≈ 7.2.

Since Bob goes first, the total expected number of turns until the game is over is the sum of the expected number of turns for Bob, Sally, and Linda, which is 9 + 7.2 + 7.2 = 23.4 turns.

Answer 2

To compute the expected number of turns until the game is over, we calculate the probabilities of each player winning in a turn and then add the reciprocals of these probabilities. The expected number of turns is 23.4.

Step-by-step explanation:

To compute the expected number of turns until the game is over, we need to calculate the probability of each player winning in a given turn. Let's start with Bob:

Bob can obtain a sum of 5 by rolling a 1 and a 4, or a 2 and a 3, or a 3 and a 2, or a 4 and a 1. So, there are 4 favorable outcomes out of 36 possible outcomes, since each of the two dice has 6 faces.

The probability of Bob winning in a turn is therefore 4/36, which simplifies to 1/9.

Similarly, we can calculate the probabilities of Sally and Linda winning in a turn. For Sally, the probability is 5/36, and for Linda, it's 5/36.

Now, to find the expected number of turns until the game is over, we add the reciprocals of these probabilities:

Expected number of turns = 1/(1/9) + 1/(5/36) + 1/(5/36) = 9 + 7.2 + 7.2 = 23.4 turns.