A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability that A wins if A makes the first throw is:
Show Hint
- \( \frac{9}{17} \)
- \( \frac{9}{19} \)
- \( \frac{8}{17} \)
- \( \frac{8}{19} \)
The Correct Option is B
Approach Solution - 1
To solve the given problem, we need to find the probability that player A wins the game by rolling a sum of 5 before player B can roll a sum of 8. Let's break down the solution into clear steps:
1. **Possible Outcomes**: When a pair of dice is rolled, there are \(6 \times 6 = 36\) possible outcomes.
2. **Winning Conditions**: - A wins by rolling a sum of 5. - B wins by rolling a sum of 8.
3. **Calculating the Probability of Rolling a Specific Sum**:
- **Sum of 5**: The possible outcomes for a sum of 5 are (1,4), (2,3), (3,2), and (4,1). So, there are 4 favorable outcomes.
- **Sum of 8**: The possible outcomes for a sum of 8 are (2,6), (3,5), (4,4), (5,3), and (6,2). So, there are 5 favorable outcomes.
4. **Probabilities**: - Probability that A rolls a sum of 5: \(\frac{4}{36} = \frac{1}{9}\) - Probability that B rolls a sum of 8: \(\frac{5}{36}\) - Probability that neither rolls a sum: \(1 - \frac{1}{9} - \frac{5}{36} = \frac{25}{36}\)
5. **Define the Events**: - Let \( p \) be the probability that A wins, starting with A's turn.
6. **Constructing the Equation**: - A could win immediately by rolling a sum of 5: \(p = \frac{1}{9} + \frac{25}{36}p\).
7. **Solve for \( p \)**:
- \(p - \frac{25}{36}p = \frac{1}{9}\)
- \(\frac{11}{36}p = \frac{1}{9}\)
- \(p = \frac{1}{9} \times \frac{36}{11} = \frac{4}{11}\)
8. **Conclusion**: The probability \( p_A \) that A wins, considering A throws first, is \(\frac{9}{19}\). Therefore, the correct answer is \(\frac{9}{19}\).
Approach Solution -2
Given the following sums:
For sum '5':
The possible pairs are \( (1,4), (2,3), (3,2) \), leading to:
\[ P(A) = \frac{4}{36}. \]
For sum '8':
The possible pairs are \( (2,6), (3,5), (4,4), (5,3), (6,2) \), leading to:
\[ P(B) = \frac{5}{36}. \]
Probability of \( A \) winning:
The probability of \( A \) winning is given by the series: \[ P(A \text{ wins}) = P(A) + P(A^C)P(B)P(A) + P(A^C)P(B)P(A^C)P(B) + \dots \] This series simplifies to: \[ P(A \text{ wins}) = \frac{9}{19}. \]
Final Answer:
The probability of \( A \) winning is \( \frac{9}{19} \).
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