Practice Law of Large Numbers in Statistics

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

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Quick Recap

The Law of Large Numbers states that as the number of independent, identically distributed trials increases, the sample average converges to the theoretical expected value (population mean). In other words, larger samples produce more reliable estimates of the true probability or average.

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.

Showing a random 20 of 50 problems.

Example 1

medium
A casino game has expected loss \$0.10 per play. Over 1,000 plays, the gambler is down only \$30. By LLN, what tends to happen over 100,000 plays?

Example 2

medium
A slot machine has expected value โˆ’$0.05-\$0.05 per spin. Over 20000 spins, what total result is expected, and what does the LLN say about reliability?

Example 3

challenge
A die's running average is 4.04.0 after 9 rolls (sum 36). What single next roll would bring the 10-roll average exactly to the expected 3.53.5?

Example 4

medium
Does the LLN say the sample average will exactly equal the true mean for large nn?

Example 5

easy
A roulette bet pays out with expected value โˆ’$0.05-\$0.05 per dollar. After 10,000 $1 bets, what total loss is expected?

Example 6

medium
You roll a fair die 600 times. By LLN, what is the approximate expected number of 66s?

Example 7

easy
True or false: the Law of Large Numbers guarantees that after many heads, tails becomes more likely.

Example 8

easy
A simulation's running average of a die roll is plotted as trials increase. What value does the curve approach?

Example 9

medium
In symbols, the Weak Law of Large Numbers says Xห‰nโ†’?\bar{X}_n \to ? in probability as nโ†’โˆžn \to \infty.

Example 10

medium
You simulate 10,000 rolls of a fair die. You compute the average. By LLN it should be close to 3.53.5. Approximately what standard deviation does Xห‰10000\bar{X}_{10000} have? (Die SD โ‰ˆ1.708\approx 1.708.)

Example 11

medium
Why do insurance companies rely on the Law of Large Numbers?

Example 12

easy
Does the Law of Large Numbers apply to a single coin flip?

Example 13

easy
A four-sided die labeled 1, 2, 3, 4 is rolled. What number does the long-run average of rolls approach?

Example 14

hard
A Monte Carlo estimator for ฯ€\pi uses random points in a unit square and counts those inside the inscribed quarter circle. After nn trials, 4p^โ†’ฯ€4 \hat p \to \pi where p^\hat p is the empirical fraction. Why does this work?

Example 15

medium
A game has expected value โˆ’$0.10-\$0.10 per play. Over 5000 plays, what is the expected total result?

Example 16

easy
A die has true mean roll 3.53.5. A gambler rolls 6 times averaging 4.24.2. What does LLN say about 10000 rolls?

Example 17

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 18

hard
A Cauchy-distributed random variable has no finite mean. Does the Law of Large Numbers apply to averages of i.i.d. Cauchy samples?

Example 19

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 20

easy
After 4 rolls of a fair die you got an average of 5.05.0. Does the LLN say the next 4 rolls will average 2.02.0 to balance things out?
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