# What is an example of the law of large numbers?

Feb 23, 2015

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 $\rightarrow$ casino wins/losses
1 $\rightarrow$ 1/0
10 $\rightarrow$ 4/6
100 $\rightarrow$ 62/38
1000 $\rightarrow$ 498/502
10000 $\rightarrow$ 5381/4619
100000 $\rightarrow$ 52370/48730

While in the short run (note the example with 10 spins), the casino might lose, in the long run profitability is guaranteed.