Sunday Times Teaser 3240 – The Case of the Green Cane of Kabul
by Michael Fletcher
Published Sunday October 27 2024 (link)
Inspector Lestrade had a warrant for the arrest of Dr Watson.
“Pray explain,” cried Dr Watson. “A man was badly beaten earlier tonight near here. The victim told us that the assailant used a green cane. We all know that you received the Green Cane of Kabul for military service and are the only person allowed to carry it.”
“What you say is true, Lestrade”, said Holmes. “However, there are several recipients of the Blue Cane of Kabul living here, and 20 per cent of people seeing blue identify it as green while 20 per cent of people seeing green identify it as blue. It follows that the probability that the colour of the cane used in this attack is blue is 2/3.
How many recipients of the Blue Cane of Kabul are there in the city?
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Let \(P_{bg}\) be the probability of a blue cane seen as green and \(P_{gg}\) be the probability of a green cane seen as green. Hence if there are \(b\) blue canes in the city: \[P_{bg} = \left(\frac{1}{5}\right)\left(\frac{b}{b+1}\right)\] \[P_{gg} = \left(\frac{4}{5}\right)\left(\frac{1}{b + 1}\right)\]
We are told that the probability that a blue cane is used in the attack is 2/3, which means that a blue cane is twice as likely as a green cane. Hence: \[\left(\frac{1}{5}\right)\left(\frac{b}{b+1}\right) = 2\left(\frac{4}{5}\right)\left(\frac{1}{b + 1}\right)\]
which simplifies to show that there are 8 holders of blue canes in the city.