Discussion
Two policemen, A and B, fire once each at the same time at an escaping convict. The probability that A hits the convict is three times the probability that B hits the convict. If the probability of the convict not getting injured is 0.5, the probability that B hits the convict is
Answer: A
The probability that B hits is about 0.14.
Let B hit with probability p, so A hits with probability 3p. The shots are independent, and the convict escapes injury only if both miss:
\[ \begin{aligned} (1 - 3p)(1 - p) &= 0.5 \\ 3p^{2} - 4p + 0.5 &= 0 \\ p &= \frac{4 - \sqrt{10}}{6} \approx 0.14 \end{aligned} \]
The other root, about 1.19, is not a probability. Check: A hits with 0.42, and \(0.58 \times 0.86 \approx 0.50\). The value 0.33 would make A's probability 0.99, and 0.22 or 0.40 give a non-injury probability far from 0.5.
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