Page 86 - Probability Demystified

CHAPTER 4 The Multiplication Rules 75

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tenth toss is , and the probability of getting a tail on the tenth toss is ,so
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it doesn’t really matter since the probabilities are the same. A coin is an
inanimate object. It does not have a memory. It doesn’t know that in the
long run, the number of heads and the number of tails should balance out.
So does that make the law of averages wrong? No. You see, there’s a big
difference between asking the question, ‘‘What is the probability of getting
10 heads if I toss a coin ten times?’’ and ‘‘If I get 9 heads in a row, what is
the probability of getting a head on the tenth toss?’’ The answer to the first
question is 1 10 ¼ 1 , that is about 1 chance in 1000, and the answer to the
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second question is 1 .
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This reasoning can be applied to many situations. For example, suppose
that a prize is offered for tossing a coin and getting 10 heads in a row. If you
played the game, you would have only one chance in 1024 of winning, but if
1024 people played the game, there is a pretty good chance that somebody
would win the prize. If 2028 people played the game, there would be a good
chance that two people might win. So what does this mean? It means that the
probability of winning big in a lottery or on a slot machine is very small, but
since there are many, many people playing, somebody will probably win;
however, your chances of winning big are very small.
A similar situation occurs when couples have children. Suppose a husband
and wife have four boys and would like to have a girl. It is incorrect to reason
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that the chance of having a family of 5 boys is , so it is more likely that the
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next child will be a girl. However, after each child is born, the probability that
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the next child is a girl (or a boy for that matter) is about . The law of
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averages is not appropriate here.
My wife’s aunt had seven girls before the first boy was born. Also, in the
Life Science Library’s book entitled Mathematics, there is a photograph of
the Landon family of Harrison, Tennessee, that shows Mr. and Mrs. Emery
Landon and their 13 boys!
Another area where people incorrectly apply the law of averages is in
attempting to apply a betting system to gambling games. One such system is
doubling your bet when you lose. Consider a game where a coin is tossed. If it
lands heads, you win what you bet. If it lands tails, you lose. Now if you bet
one dollar on the first toss and get a head, you win one dollar. If you get tails,
you lose one dollar and bet two dollars on the next toss. If you win, you are
one dollar ahead because you lost one dollar on the first bet but won two
dollars on the second bet. If you get a tail on the second toss, you bet four
dollars on the third toss. If you win, you start over with a one dollar bet, but
if you lose, you bet eight dollars on the next toss. With this system, you win
every time you get a head. Sounds pretty good, doesn’t it?