Practice Law of Large Numbers in Statistics

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

As the number of trials increases, the experimental probability (sample average) converges to the theoretical probability (population mean).

Flip a coin 10 times: maybe 7 heads (70%). Flip 100 times: closer to 50%. Flip 10,000 times: very close to 50%. More trials = more reliable averages. Short-run luck evens out.

Example 1

easy
A coin is flipped and the running proportion of heads is recorded: After 10 flips: 0.70, after 50: 0.56, after 200: 0.52, after 1000: 0.498, after 10,000: 0.5012. What does this pattern illustrate?

Example 2

medium
A casino offers a game where you win \1 with probability 0.48 and lose \1 with probability 0.52. (a) What is the expected value per game? (b) Explain why the casino is guaranteed to profit in the long run using the law of large numbers.

Example 3

medium
A basketball player has a career free-throw percentage of 80%. She misses 5 free throws in a row in one game. A commentator says she is 'due' to make the next one. Is this reasoning correct? Explain using the law of large numbers.

Example 4

hard
An insurance company insures 100,000 homeowners. The probability of a claim in a given year is 0.02, with an average claim of \50,000. They charge \1,200 per policy. (a) What is the expected number of claims? (b) What is the expected total payout? (c) What is the expected profit? (d) Explain why this business model works using the law of large numbers.
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