Mary Reyes

2022-01-17

What is an example of the law of large numbers?

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Expert

Example 1:
Suppose the population within a certain area is such that there are 3,497,419 females and 2,812,822 males (for a total population of 6,310,241)
If a random sample of 12 jurors is selected from this region, how many of the jurors will be women?
For an individual case, the answer could be anywhere from 0 to 12.
The law of large numbers says that if this process is repeated many times the average will be
$12\cdot \frac{3497419}{6310241}$
=6.65 (approximately)
Example 2:
A casino runs a wheel with numbers from 0 to 20.
A player pays $1 to play for each spin and gets$2 back if the wheel comes up with an odd number.
Since there are 11 odd numbers, the casino could be expected to win 11 out of 21 times (roughly 52.4% of the time).
With these values in mind a simulation was run with varying numbers of spins with the following results:
number of spins $\to$ casino wins/losses
$1\to \frac{1}{0}$
$10\to \frac{4}{6}$
$100\to \frac{62}{38}$
$1000\to \frac{498}{502}$
$10000\to \frac{5381}{4619}$
$100000\to \frac{52370}{48730}$
While in the short run (note the example with 10 spins), the casino might lose, in the long run profitability is guaranteed.

William Appel

Expert