During each turn of a game, a player rolls fair six-sided die with sides numbered 1 through 6 and adds...
GMAT Two Part Analysis : (TPA) Questions
During each turn of a game, a player rolls fair six-sided die with sides numbered 1 through 6 and adds the result to his or her cumulative score in the game. Then, next player takes a turn. The first player to get a cumulative score of 20 points or more wins. Torrance and Adia are playing the game. There are no other players. Torrance has 15 points and Adia has 17. It is Torrance's turn and he is about to roll the dice. Let \(\mathrm{X}\) equal the probability that Torrance will win the game during his current turn.
Select for First and for Second the two operations such that applying the first to \(x\) and then applying the second to the result of the first would yield the probability that Adia will win on her next turn. Make only two selections, one in each column.
Let me work through this step-by-step.
First, I need to find X, the probability that Torrance wins on his current turn.
Torrance has 15 points and needs 20+ to win, so he needs to roll 5 or more. On a fair six-sided die, he can roll 5 or 6 to win immediately. \(\mathrm{P(rolling\ 5\ or\ 6)} = \frac{2}{6} = \frac{1}{3}\) Therefore, \(\mathrm{X} = \frac{1}{3}\)
Next, I need to find the probability that Adia wins on her next turn.
For Adia to get her next turn, Torrance must not win on his turn. \(\mathrm{P(Torrance\ doesn't\ win)} = 1 - \mathrm{X} = 1 - \frac{1}{3} = \frac{2}{3}\)
Adia has 17 points and needs 20+ to win, so she needs to roll 3 or more. \(\mathrm{P(Adia\ rolls\ 3,\ 4,\ 5,\ or\ 6)} = \frac{4}{6} = \frac{2}{3}\)
The probability that Adia wins on her next turn is: \(\mathrm{P(Torrance\ doesn't\ win)} \times \mathrm{P(Adia\ wins)} = (1 - \mathrm{X}) \times \frac{2}{3}\)
Starting with \(\mathrm{X} = \frac{1}{3}\): First operation: Subtract from 1 → \(1 - \frac{1}{3} = \frac{2}{3}\) Second operation: Multiply by 2/3 → \(\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}\)
Therefore, the two operations are:
- Subtract from 1
- Multiply by 2/3