# Questionbank: Permutation and Probability

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## Probability Duel

Q.14: 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?
1. ${7 \over 10}$
2. ${12 \over 23}$
3. ${2 \over 3}$
4. ${11 \over 17}$

Choice A. 11

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## Detailed Solution

A can win in the following scenarios:
A strikes in first shot.

A misses in the first shot, B misses in the second, A strikes in the third.

A misses in the 1st shot, B misses in the 2nd, A misses the 3rd, B misses the 4th and A strikes the 5th.

And so on…

P(A in 1st shot) = 0.6
P(A in 3rd shot) = 0.4 × 0.2 × 0.6 {A misses, then B misses and then A strikes}
P(A in 5st shot) = 0.4 × 0.2 × 0.4 × 0.2 × 0.6 {A misses, then B misses and then A misses, then B misses, then A strikes}

Overall probability = Sum of all these
= 0.6 + 0.4 × 0.2 × 0.6 + 0.4 × 0.2 × 0.4 × 0.2 × 0.6 + …

Which is nothing but
0.6 + (0.4 × 0.2) × 0.6 + (0.4 × 0.2)2 × 0.6 + (0.4 × 0.2)3 × 0.6…

This is an infinite geometric progression with first term 0.6 and common ratio 0.4 × 0.2
Required Probability = ${a \over 1 - r}$ = ${0.6 \over 1 - 0.08}$ = ${0.6 \over 0.92}$ = ${30 \over 46}$ = ${15 \over 23}$

Correct Answer : ${15 \over 23}$

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