Marilyn’s Coins

The last several postings to this site have involved the huge changes in probabilities that arise when events are coupled in non-obvious ways. The examples included a three-dice gambling game and the collapse of the world-wide economic system. These puzzles are not pie-in-the-sky exercises, but have real world implications.

So I was pleasantly surprised when this Sunday in the Parade magazine, the “Ask Marilyn” column featured another example similar to one I posted. Since you know the thesis is that the actual probability is different from what you initially assume, you will likely get the answer quickly, but try it on friends and see how they respond. Here is my version of Marilyn’s puzzle:

A street huckster shows you a bag with two identical coins in it. At least they look identical. He then shows you that one of them is normal, but the other has two heads and no tails. He puts the coins in the bag and shakes them. He pulls one coin out and lays it on his portable table with heads up. He then offers you even odds the other side is heads. That is, you are betting the hidden side is tails. Do you take the bet?

A reasonable way of looking at this proposed wager is that if both coins were placed on the table with heads showing up, then one of them would have tails on the hidden side and the other one would have heads (the two-headed coin). Therefore the chances are even that either coin could have tails on the don side, and the bet is fair.

After gambling several times, you are somewhat poorer, and the huckster leaves suddenly when a police cruiser glides by.

What happened?

The correct analysis is similar to the boy-girl puzzle I presented earlier and the Monty Hall paradox. For convenience, label the heads H1 and H2 for the two-headed coin. Assume H3 is on the normal coin. The tails need no label. When showing any one side of either coin, one of the four possible sides is shown, but we are told the up side is a heads. That eliminates one possibility. The visible side could be any of the three heads. First possibility is that the showing face is H1; then the bottom is H2. The second possibility is that the showing face is H2; then the bottom is H1. The third possibility is that the showing face is H3; then the bottom is tails. Of the three possible choices, two of them have you losing, and only one lets you win. This is not a fair bet.

Again what at first seems to be two independent events turns out to be linked is a way that changes the predicted outcome substantially. If you are a bit unscrupulous, you could try this on gullible friends using some ad hoc substitute for a two-headed coin. For instance, take two identical poker chips and mark one side on one on them with a pen. That will serve as the tails side. The unmarked three sides serve as the heads. You can even start the wager by describing the false analysis given above to prejudice your victim’s thought process. Good luck! (Although luck is not really necessary, is it?)