Practice Law of Large Numbers (Intuition) in Math

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 trials increases, the sample mean converges to the true population mean โ€” randomness averages out over many repetitions.

As the number of trials grows, the sample mean converges to the true expected value โ€” randomness averages out over many trials, making the average predictable.

Showing a random 20 of 50 problems.

Example 1

easy
A fair coin is flipped 10,000 times. The proportion of heads will be close to what value?

Example 2

easy
A fair die's average roll over many throws approaches what number?

Example 3

easy
Does the LLN guarantee anything about small samples?

Example 4

easy
A die is rolled repeatedly and the running average is tracked. After 5 rolls, the average is 3.8. After 500 rolls, the average is 3.52. What does the LLN predict will happen to the average as rolls โ†’ โˆž?

Example 5

medium
Rolling two dice, the average of the sum over 10,000 rolls approaches what value?

Example 6

challenge
A fair coin shows 520 heads in 1000 flips (0.520). To dilute this 0.020 surplus so the running proportion is within 0.005 of 0.50, roughly how many additional fair flips are needed (assume they land near 0.50)?

Example 7

medium
A simulation estimates ฯ€\pi by random points; with more points the estimate stabilizes near 3.14159. Which law explains this convergence?

Example 8

challenge
To halve the standard error of a sample mean (which scales as 1/n1/\sqrt{n}), by what factor must nn increase?

Example 9

medium
A weather app says '60% chance of rain' on many days. Over a long record, on what fraction of such days should it actually rain if well-calibrated?

Example 10

easy
Is the LLN the same as the gambler's fallacy?

Example 11

medium
Insurance: one policyholder's loss is unpredictable, but an insurer with 1,000,000 policies can predict total payouts well. Why?

Example 12

medium
An insurance company writes 10,00010{,}000 identical home policies with expected payout $300\$300 each. By LLN, what total payout does it expect?

Example 13

easy
A fair coin is flipped. Show how the proportion of heads approaches 0.5 as nn increases, using simulation results: n=10n=10: 6 heads; n=100n=100: 53 heads; n=1000n=1000: 498 heads.

Example 14

medium
A pollster surveys 2525 voters in a state of 55 million. Why does this small sample produce a noisy estimate of voter preference?

Example 15

hard
A roulette wheel has 1818 red, 1818 black, 22 green pockets. A player bets $1\$1 on red each spin (pays $1\$1 if red, โˆ’$1-\$1 otherwise). Find expected loss per spin, and the expected total loss over 10001000 spins.

Example 16

medium
A weather model is 'well-calibrated' if on days it says '70% chance of rain', it rains about 70%70\% of the time long run. Which law underpins this?

Example 17

easy
To estimate a coin's true bias, is it better to flip it 20 times or 5,000 times?

Example 18

easy
A coin shows 77 heads in 1010 flips, 5151 heads in 100100 flips, and 5,0035{,}003 heads in 10,00010{,}000 flips. State each proportion and which is closest to the truth.

Example 19

easy
Does the law of large numbers guarantee that 100 flips give exactly 50 heads?

Example 20

hard
Strong vs Weak LLN: state the difference in one sentence each.
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