## Two more fun fallacies

Here are two more well-known problems that show how easy it is to get confused by formulation or by a persuasive argument.

1. The missing pound. Three men rent a room in a hotel. The owner tells them the price is £30, so each gives him a tenner. Later the owner realises that there is a special deal on and that in fact the price is £25. He sends a bellboy to the room with five one pound coins. The bellboy gives the coins to the men, who each take a pound and give the bellboy a £2 tip.

The problem is this: they each paid £9, which makes a total of £27. Adding the £2 tip will make it £29. Where has the missing pound gone?

2. The curious number of heads. This is a bit more serious. Suppose a friend of yours is holding a coin in each fist. She says, One of my coins is heads up.  What is the probability that the other coin is tails up? Here is an argument that says that it is $2/3$: since the possible outcomes are TT, HT (i.e. heads in the left hand and tails in the right one), TH and HH all with equal probability, and we know that TT is impossible, it means that the probability of {HT or  TH} is $2/3$.

Is this argument correct?  It certainly has convinced many bright people…

(MG)