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by Allan N Wilson
A general idea that many people hold goes something like this:
The longer a series of chance events goes on, the more likely it is that
things will "even up". For example, the longer you drive a car, the more
likely you are to have your share of flat tyres. The more children you have,
the more likely you will have an equal division of boys and girls (assuming
an even number of children). The longer you flip a coin, the more likely
the number of heads and tails will equalise.
Actually, the latter two examples are identical, if certain simple assumptions
are made. In the case of the coin, we shall suppose that the coin is "fair",
so that on any one flip, head and tail are equally likely. In the case of
the children, we shall rule out twins or other multiple births, and shall
assume that the parents do not have any tendency to produce children of
one sex as preferred to the other. Of course there is the practical consideration
that one can flip a coin a lot more times than a woman can bear a child.
So, it is more natural to use the coin for purposes of illustrating large
samples. But for small samples, there is something unique about each child,
which makes the family a marvellous illustration for stressing certain points.
3.1 The Law of Large Numbers
I have never actually seen a statement of a law of averages as such in any
book on probability or statistics. However, I have seen references to a
law of large numbers, which in rather loose language could be expressed
this way: the greater the number of trials, the smaller will be percentage
fluctuations away from the expected (or average) number of successes, but
the larger will be the absolute fluctuations. Note carefully that this deals
with fluctuations about the expected number of successes, and not with the
expected number itself.
Let's see how this applies to our example of children in the family (or
to the flipping of a coin). We shall consider the various boy-girl combinations
that can occur in successively larger families, starting with two children,
and ranging on up through 4, 6, 8, 10, 100, 10,000 and 1,000,000. In the
last three cases, we shall, of course, switch over mentally from children
to coins. We shall find these examples very instructive.
At each birth, there are two possibilities for the sex of the child, B (boy)
or G (girl). In a two child family, therefore, there are 2 x 2 = 4 possible
orders in which the two children are born. These are indicated in the following
table.
DISTRIBUTION OF BOY-GIRL BIRTHS IN TWO-CHILD FAMILY
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BOTH BOYS
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EQUAL DIVISION
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BOTH GIRLS
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B-B
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B-G/G-B
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G-G
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| CHANCE |
1/4
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2/4
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1/4
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AS A %
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25%
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50%
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25%
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Now we move on to the four child family. There are 2 x 2 x 2 x 2 = 16 possible
orders. (A more compact way to represent this is 2-4 which means 2 to the
power 4, or the produce of four 2s). These 16 possibilities tabulate as
in the following table.
DISTRIBUTION OF B-G IN FOUR CHILD FAMILY
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4B
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3B and 1G
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2B and 2G
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1B and 3G
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4G
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B-B-B-B
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B-B-B-G
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B-B-G-G
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B-G-G-G
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G-G-G-G
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B-B-G-B
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B-G-B-G
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G-B-G-G
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B-G-B-G
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B-G-G-B
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G-G-B-G
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G-B-B-B
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G-B-B-G
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G-G-G-B
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G-B-G-B
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G-G-B-B
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1/16
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4/16
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6/16
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4/16
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1/16
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6 ¼%
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25%
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37½%
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25%
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6 ¼%
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Notice that in the four child family, there is less chance of an exactl
Date Posted: 30-May-1999
Excerpt from "The Casino Gamblers Guide" by Allan N Wilson
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