Short runs wander
The theoretical probability of heads on a fair coin is , which is 0.5. In only 10 throws, though, 7 heads is not unusual. That is an experimental probability of , a long way from 0.5. In a short run, a few extra heads pull the fraction far off.
Long runs settle
Keep throwing, and keep a running count. In one experiment, 70% of the throws were heads after 10 throws, 58% after 50 throws, 53% after 200 throws and 51% after 1000 throws. The more throws there were, the closer the share of heads came to half.
The percentage of the throws that were heads, after 10, 50, 200 and 1000 throws. The line drops toward 50% as the throws add up.
The running count after 10, 50, 200 and 1000 throws, one row each. In each row the colored part is the share of heads and the rest is the share of tails. The dashed line is half: after 10 throws the heads reach well past it, and after 1000 throws they stop just past it.
The law of large numbers
This pattern has a name, the law of large numbers: the more trials there are, the closer the experimental probability usually comes to the theoretical probability. 10 throws can land far from , and 1000 throws rarely do.
So an experimental probability from many trials is a better estimate than one from a few trials. 7 heads in 10 throws is no reason to call a coin unfair, but 700 heads in 1000 throws would be.
With more and more throws, the experimental probability of heads settles at .
The coin does not remember
The law does not say that a run of heads makes tails due. After five heads in a row, the next throw is still heads with probability . The coin has no memory: each throw is independent, which means that no throw changes the chances on any other.
So how does the share of heads settle? The early extra heads are not taken back. They are outweighed. Say the first 10 throws gave 7 heads, 2 more than half. If the next 990 throws come up heads about half the time, that is about 495 more heads, and 7 + 495 = 502 heads in 1000 throws, which is 0.502. The 2 extra heads are still there, but now they are 2 out of 1000 instead of 2 out of 10.
Worked example: A Class Spinner, and Its Greens After 10, 100 and 1000 Spins
Question A spinner has 4 equal sections colored green, red, blue and yellow. A class spins it many times and keeps a running total of the greens. After 10 spins there are 5 greens, after 100 spins there are 29 greens, and after 1000 spins there are 256 greens. (a) What is the theoretical probability of green, as a decimal? What is the experimental probability of green after the first 10 spins? (b) What is the experimental probability of green after all 1000 spins, and how far is it from the theoretical probability?
1.1 of the 4 equal sections is green, so the theoretical probability of green is 14 = 0.25.
One of the 4 equal sections is green: 14 = 0.25 in theory. 2.(a) After 10 spins, the experimental probability is 510 = 0.5. That is twice the theoretical probability. In only 10 spins, a gap this big can happen.
(a) After 10 spins: 510 = 0.5, twice the theoretical probability. 3.After 100 spins, it is 29100 = 0.29, which is 0.29 − 0.25 = 0.04 away from 0.25.
After 100 spins: 29100 = 0.29, which is 0.04 from 0.25. 4.(b) After 1000 spins, it is 2561000 = 0.256, which is only 0.256 − 0.25 = 0.006 away from the theoretical probability.
(b) After 1000 spins: 2561000 = 0.256, only 0.006 from 0.25. 5.The gaps shrink from 0.25 to 0.04 to 0.006. With more spins, the experimental probability settles close to 0.25.
The gaps shrink as the spins grow: the experimental probability settles near 0.25.
Answer: (a) 0.25 in theory, and 0.5 after 10 spins; (b) 0.256, which is 0.006 away from 0.25
Common mistakes
- Deciding after 10 spins that the spinner is unfair because green came up half the time. Ten spins are too few: a fair spinner often gives results far from 14 in a short run.
- Expecting exactly 250 greens in 1000 spins. The law of large numbers says the fraction of greens comes close to 14, not that the count lands exactly on 250.