## Those two fun fallacies exploded

Here’s how to resolve the two fallacies we presented back in December…

1. The problem of the missing pound is interesting in that there is no problem! Consider: suppose the price of the room were £10. The three men paid the owner £30, so he would give the bellboy 20 pound coins. The bellboy would then give each man £6 and would pocket a £2 tip. Does it make sense to say then: each man paid £4, plus the £2 tip, so where have £16 gone? There is no need to add the tip to what was paid: you have instead to subtract it from what was paid, and you should get the price of the room. So originally the 3 men paid £9 each, i.e. £27 all together; subtract from that the £2 tip, and bingo, you have the £25 room-worth, and not a penny missing from anyone’s pocket.

2. This argument is incorrect. I have heard that it convinced Lewis Carroll, which is either scandalous or a scandalous libel (see how neatly I cover all exits?). The assumption here is that the new information, that at least one coin is heads (H), influences equally all outcomes that are still possible (HH, TH, HT) since TT is no longer available (as is the case in the Monty Hall problem).

Let ${\mathbb P}(A)$ denote the probability of event $A$. Then (easier in words): ${\mathbb P}(HH)$ is equal to the sum of the probabilities of the following two (mutually exclusive!) events: that the heads of which we have been informed is in the right hand and the other coin is heads, and that the heads of which we have been informed is in the left hand and that the other coin is heads. The first event has probability $1/4$ as does the second one, so ${\mathbb P}(HH)$ is $1/2$ and not $1/3$. Clearly, ${\mathbb P}(HT)= {\mathbb P}(HT) =1/4$, so the Gods of Probability are appeased.

(MG)