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
