What is an example of the law of large numbers?

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ol3i4c5s4hr · Accepted Answer

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 · Accepted Answer

As more trials are performed, probability approaches classical probability. For example, if a fair coin is tossed 1,000,000 times, about half of the tosses will come up heads, and half will come up tails.

What is an example of the law of large numbers?

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