Chapter 18 Deviation from the Mean644
For example, given any set of mutually independent events, if you expect 10 of
them to happen, then at least one of them will happen with probability at least 1
e ^10. The probability that none of them happen is at moste ^10 < 1=22000.
So if there are a lot of independent things that can go wrong and their probabil-
ities sum to a number much greater than 1, then Theorem 18.7.4 proves that some
of them surely will go wrong.
This result can help to explain “coincidences,” “miracles,” and crazy events that
seem to have been very unlikely to happen. Such events do happen, in part, because
there are so many possible unlikely events that the sum of their probabilities is
greater than one. For example, someonedoeswin the lottery.
In fact, if there are 100,000 random tickets in Pick-4, Theorem 18.7.4 says that
the probability that there is no winner is less thane ^10 < 1=22000. More generally,
there are literally millions of one-in-a-million possible events and so some of them
will surely occur.
18.8 Really Great Expectations
Making independent tosses of a fair coin until some desired pattern comes up is a
simple process you should feel solidly in command of by now, right? So how about
a bet about the simplest such process —tossing until a head comes up? Ok, you’re
wary of betting with us, but how about this: we’ll letyou set the odds.
18.8.1 Repeating Yourself
Here’s the bet: you make independent tosses of a fair coin until a head comes up.
Then you will repeat the process. If a second head comes up in the same or fewer
tosses than the first, you have to start over yet again. You keep starting over until
you finally toss a run of tails longer than your first one. The payment rules are that
you will pay me 1 cent each time you start over. When you win by finally getting a
run of tails longer than your first one, I will pay you some generous amount. And
by the way, you’re certain to win —whatever your initial run of tails happened to
be, a longer run will occur again with probability 1!
For example, if your first tosses areTTTH, then you will keep tossing until you
get a run of 4 tails. So your winning flips might be
TTTHTHTTHHTTHTHTTTHTHHHTTTT:
In this run there are 10 heads, which means you had to start over 9 times. So you
would have paid me 9 cents by the time you finally won by tossing 4 tails. Now
