To solve the problem, we need to determine the probability that Player A wins the sequence and the probability that Player B wins the sequence, given their respective winning conditions:

• Player A wins if he rolls a 6 on his turn.
• Player B wins if he rolls a 5 or 6 on his turn, but only if Player A does not win in the preceding round.

Let's define the following probabilities:

• $P(A)$: Probability that Player A wins.
• $P(B)$: Probability that Player B wins.

Step-by-Step Solution:

1. Probability of Player A winning on his first roll:

   • Player A rolls a 6: $P(\text{A wins on first roll}) = \frac{1}{6}$.

2. Probability that Player A does not win on his first roll:

   • Player A does not roll a 6: $P(\text{A does not win}) = \frac{5}{6}$.

3. Probability of Player B winning on his first roll:

   • Player B rolls a 5 or 6: $P(\text{B wins on first roll}) = \frac{2}{6} = \frac{1}{3}$.

4. Probability that Player B does not win on his first roll:

   • Player B rolls something other than 5 or 6: $P(\text{B does not win}) = \frac{4}{6} = \frac{2}{3}$.

Recursive Probability Calculations:

Let $P(A)$ and $P(B)$ denote the overall probabilities of Player A and Player B winning the sequence, respectively.

Player A's Probability:

• Player A can win immediately on his first roll with probability $\frac{1}{6}$.
• If Player A doesn't win on the first roll, Player B has a chance to win. If Player B doesn't win, the sequence resets to the original state (with Player A rolling first again).

Thus, the probability that Player A eventually wins is: $P(A) = \frac{1}{6} + \left(\frac{5}{6}\right)\left(\frac{2}{3}\right)P(A)$

Simplifying this equation: $P(A) = \frac{1}{6} + \frac{10}{18}P(A)$ $P(A) - \frac{10}{18}P(A) = \frac{1}{6}$ $\frac{8}{18}P(A) = \frac{1}{6}$ $P(A) = \frac{1}{6} \times \frac{18}{8} = \frac{3}{8}$

Player B's Probability:

• Since the probabilities must sum to 1 (either Player A or Player B must eventually win), we have: $P(B) = 1 - P(A) = 1 - \frac{3}{8} = \frac{5}{8}$

Final Probabilities:

• Probability that Player A wins the sequence: $\frac{3}{8}$.
• Probability that Player B wins the sequence: $\frac{5}{8}$.

Would you like further clarification on any part of this solution, or do you have any other questions?

Here are 5 related questions to consider:

1. What would be the probabilities if Player A had to roll a 5 or 6 to win?
2. How would the game change if Player B could only win by rolling a 6?
3. What is the expected number of rounds until a player wins?
4. How does the probability change if Player B rolls first instead of Player A?
5. What is the probability of each player winning if Player A needs a 6, but Player B needs a 4, 5, or 6 to win?

Tip: Recursive probabilities often arise in games where outcomes depend on the results of repeated trials. Solving such problems typically involves recognizing the recursive structure and setting up the appropriate equation.