A and B take part in a duel. A can strike with an accuracy of 0.6. B can…
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A and B take part in a duel. A can strike with an accuracy of 0.6. B can strike with an accuracy of 0.8. A has the first shot, post which they strike alternately. What is the probability that A wins the duel?
- A.
7/10
- B.
15/23
- C.
2/3
- D.
11/17
Attempted by 1 students.
Show answer & explanation
Correct answer: B
Key idea: A can win on his 1st, 3rd, 5th, … shots. Each time A and B both miss a full pair of turns (A misses then B misses), the situation resets.
Probability A wins on his 1st shot = 0.6
Probability A wins on his 3rd shot = (A misses)×(B misses)×(A hits) = 0.4×0.2×0.6 = 0.08×0.6
In general, probability A wins on his (2k+1)-th shot = (0.08)^k × 0.6, where 0.08 = 0.4×0.2
Sum these probabilities to get the total probability A wins:
Total = 0.6 × (1 + 0.08 + 0.08^2 + …) = 0.6 × 1/(1−0.08) = 0.6/0.92 = 15/23 ≈ 0.6522