Stick or roll is a game involving two players, A and B, and a die with four faces (numbered 1, 2, 3, 4). The faces are equally likely to occur when the die is rolled. Player A rolls the die once and sticks with that number as the score or rolls it again and scores the sum of the two numbers. • If A's score is greater than 4 then A loses. • If A's score is 4 then A wins. If A's score is less than 4, then B rolls the die once and sticks with that number as the score or rolls it again and scores the sum of the two numbers. • If B's score is greater than 4 then B loses. • If B's score is 4 or less and equal to A's or less, then B loses. • If B's score is 4 or less and greater than A's, then B wins. Player B sticks on the first roll if that number wins and rolls again if it doesn't win and it is possible to win with a second roll. Some
Stick or roll is a game involving two players, A and B, and a die with four faces (numbered 1, 2, 3, 4). The faces are equally likely to occur when the die is rolled. Player A rolls the die once and sticks with that number as the score or rolls it again and scores the sum of the two numbers. • If A's score is greater than 4 then A loses. • If A's score is 4 then A wins. If A's score is less than 4, then B rolls the die once and sticks with that number as the score or rolls it again and scores the sum of the two numbers. • If B's score is greater than 4 then B loses. • If B's score is 4 or less and equal to A's or less, then B loses. • If B's score is 4 or less and greater than A's, then B wins. Player B sticks on the first roll if that number wins and rolls again if it doesn't win and it is possible to win with a second roll. Some example games might be: • A rolls a 2, chooses to roll again and rolls a 3. A's score is then 5 so A loses and B wins. * A rolls a 2 and chooses not to roll again. B rolls a 2, so rolls again and rolls another 2. B's score is then 4, so B wins. • A rolls a 1, chooses to roll again and rolls a 2. B rolls a 2, rolls again and rolls a 3. B's score is then 5 so B loses and A wins. In solutions to the following questions express all probabilities exactly.
Player A decides to stick if the first roll is 3 or 4, otherwise rolls again. What is the probability that A wins the game using this strategy?
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