Puzzles
Downing Street
A knot of spectators in Downing Street was watching members of the Cabinet as they arrived for a critical meeting.
"Who's that?" I asked my neighbour, as a silk-hatted figure, carrying rolled umbrella, rang the bell at No. 10. "Is it the Minister of Maths?"
"Yes," he said.
"Quite right," said a second spectator. "The Minister of Maths it is. Looks grim, doesn't he?"
The first of the speakers tells the truth three times out of four. The second tells the truth four times out of five.
What is the probability that the gentleman in question was in fact the Minister of Maths?
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The probabilities can be summarised as follows:
| First person truthful | First person lying |
| Second person truthful | \(\frac{3}{4}\times\frac{4}{5}=\frac{12}{20}\) | \(\frac{1}{4}\times\frac{4}{5}=\frac{4}{20}\) |
| Second person lying | \(\frac{3}{4}\times\frac{1}{5}=\frac{3}{20}\) | \(\frac{1}{4}\times\frac{1}{5}=\frac{1}{20}\) |
As they both agree, only both lying and both truthful are possible. Hence the chance of them lying is 1/13 and the chance of them telling the truth, and it indeed being the Minister of Maths, is 12/13
Extension
If the first person said it was the Minister of English and the second said it was the Minister of Maths, what is the probability that it was the Minister of Maths?
