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Curiosity · Chance and probability

The law of large numbers

Nobody can predict a single roll of a die, so why do a thousand rolls settle into a surprisingly steady pattern? Find out how relative frequency approaches probability as the number of trials grows, and why insurers and casinos rely on this law.

  • 16 min
  • 1 simulation
  • Mini quiz

Updated:

Preview

The question

You roll two dice. You can't know in advance whether the total will be 7. You can't know on the next roll either. But if you rolled them a thousand times, could you predict roughly how many times the total would be 7? How do events that can't be predicted one by one become predictable when you put them together?

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Probability lab

A total of 7 in a thousand rolls

First roll a few times one at a time, then try the 100 and 1,000 roll buttons. Watch how the line on the graph calms down. Make your estimate with the slider; it appears on the graph as a dashed line.

Prediction

Predict first

If you rolled two dice 1,000 times, roughly what proportion of rolls do you think would give a total of 7?

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Frequently asked

5 questions

Does the law of large numbers say that results "even out" in the long run?

No. The law does not say that deviations are made up for by later trials. Extra results get diluted among a growing number of trials; the coin does not remember the past, yet the proportion still gets close to the probability.

If an event has probability 1/6, does it happen once every 6 tries?

No. If you roll two dice 6 times, a total of 7 might not come up at all, or it might come up three times. What the probability tells you is that over many rolls the share of 7s settles near 1/6, about 16.7%.

If the proportion approaches 0.5, does the gap between heads and tails shrink too?

Not necessarily. 7 heads in 10 tosses is 2 more than the expected 5, a proportion of 70%. 520 heads in 1,000 tosses is 20 more than expected, but the proportion is 52%. The gap grew, yet the proportion still moved closer to 0.5.

Why can a small sample be misleading?

Proportions swing much more in small samples. In a hospital where 15 babies are born a day, days when more than 60% are girls happen more often than in a hospital with 45 births a day.

Why do insurance companies and casinos rely on this law?

Nobody can know whether one person will have an accident or how a single game will end. But across hundreds of thousands of policies or millions of games, the proportions come out very close to the probabilities, so the overall result can be predicted well.