The law of large numbers
Why an observed frequency approaches the probability, what the Chebyshev inequality guarantees, how many draws give what precision, why chance never catches up and how the Monte Carlo method takes advantage.
- 26 cards
- 6 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
- Concepts
- Questions
- Sort and link
- The numbers
- Scenarios
- Understand
A few cards from the deck
Law of large numbers: what is it?
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The result saying that the observed mean of n independent repetitions approaches the expected value as n grows large.
It is the mean, or the frequency, that converges, not the total: the number of heads generally drifts further and further from half the flips.
A fair coin has just landed heads 8 times in a row. What is the probability that the next flip is heads?
- 1/2
- Less than 1/2, because tails is overdue
- More than 1/2, because the coin may be rigged
- (1/2)⁹
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1/2
A fair coin has no memory: P = 1/2 whatever the past. (1/2)⁹ is the probability of 9 heads in a row before the first flip, not of the next one.
For a frequency F over n draws of probability p, P(|F − p| ≥ ε) ≤ …, which is at most ….
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For a frequency F over n draws of probability p, P(|F − p| ≥ ε) ≤ p(1 − p)/(nε²), which is at most 1/(4nε²).
p(1 − p) never exceeds 1/4 (reached at p = 1/2), which gives a bound that no longer depends on the unknown p.
Relative frequency: what is it?
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The number of times an event occurred divided by the number of trials.
Over few trials it fluctuates a lot; the law of large numbers says it settles around the probability of the event.