Tom eagerly accepts the bet. He reasons that winning and losing are equally likely. “If I lose, then I lose the value of my necktie. But if I win, then I win

*more*than the value of my necktie. Therefore this bet is ultimately to my advantage!”

Michael is eager to accept the bet as well... since he reasons in exactly the same way.

Either guy's reasoning seems sound, yet they cannot

**both**have the advantage in the bet! Where is the fault?

**Details:**For a fair bet you'd expect both guys to have a 50% chance of winning it (100% together). In the extreme case where one guy would have a 100% chance of winning, it can only follow that the other guy has 0% chance, i.e. no chance at all. So that's why they can't both have the advantage (= a chance bigger than 50%).

Solution

Worry not, this paradox can be resolved. Let's make things a bit simpler and say that the only possible necktie prices are €5 and €10. Then four equally likely outcomes are possible:~~€5 vs. €5~~- €5 vs. €10
- €10 vs. €5
~~€10 vs. €10~~

Now, if Tom's necktie is the €5 one, then he will

**win €10**. But if Tom's necktie is the €10 one, he will

**lose €10**. The same goes for Michael! They will always win or lose the larger amount. In other words, no one has the advantage at all.

So where did the reasoning go wrong? Well, in their comparison they ignore the fact that their necktie must either be the cheaper or the more expensive one. What Tom and Michael call the ‘

*value of my necktie*’ in the losing scenario, is the same amount as what they call ‘

*more than the value of my necktie*’ in the winning scenario.

After asking their wives, however, they soon found this bet had no winners at all.

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