A special set of dice
If A is older than B, and B is older than C, it follows that A is older than C. This property is
called the transitive property. Not every relation is transitive. For example if John is the
uncle of Sam and Sam is the uncle of Jeremy, it isn’t true that therefore John is the uncle
of Jeremy. That’s fairly obvious, but sometimes a relation appears as if it should be
transitive, but isn’t. And that can be the basis of a nice little trick that will eventually be
explained using simple probability.
Bring out a set of the dice, then explain the rules:
‘I have this set of four dice. We will each choose a die; in fact since I am such a generous
person, I'll let you choose first. We’ll roll the two dice and the winner is the person whose
die has the highest number. The first person to record five wins is the champion. Now if I
am the champion, you have to do an extra hour of homework tonight. If you are the
champion, then you are excused from homework for one week. Any takers? After all,
you’ve got to be in it to win it!’
Of course, the student will be excused from doing the extra homework if the class can
figure out why the teacher wins almost all of the time.
The solution
Here are three sets of dice, each with the property that A beats B, B beats C, C beats D
and D beats A! (‘beats’ means than the probability of winning is greater than one-half).
For the first two sets of dice the probability of A beating B (and similarly for the other pairs)
is 2/3. In the last set the probability is 11/17, or .647.
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