Law of Large Numbers Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Law of Large Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept 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.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Law of Large Numbers starts by naming the possible outcomes and the event rule before assigning or combining probabilities.

Common stuck point: Students often know a procedure related to law of large numbers but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?

Worked Examples

Example 1

easy

After 4 rolls of a fair die you got an average of

5.0

. Does the LLN say the next 4 rolls will average

2.0

to balance things out?

Answer

No

First step

LLN says the long-run average converges; it does NOT predict 'balancing'.

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Example 2

medium

Bob says: 'I flipped 8 tails in a row, so the next flip is more likely heads by LLN.' Is Bob right?

Example 3

hard

A startup's daily revenue is i.i.d. with mean $5,000 and SD $2,000. By LLN, what does its average daily revenue over a year (365 days) approximately equal, and how much does that average typically vary?

Example 4

hard

For an unbiased Monte Carlo estimator

\hat\theta_n

of a parameter

\theta

with finite variance, how many trials

n

are needed to halve the typical error in the estimate?

Example 5

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 6

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A fair coin is flipped 10 times giving 7 heads. As flips increase to 10000, what fraction of heads is expected?

Example 2

easy

After 10 spins a wheel shows 'win'

60\%

of the time, but its true win probability is

0.4

. What does the LLN predict for many spins?

Example 3

easy

A die has true mean roll

3.5

. A gambler rolls 6 times averaging

4.2

. What does LLN say about 10000 rolls?

Example 4

easy

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

Example 5

easy

Which sample gives a more reliable estimate of a true probability: 50 trials or 5000 trials?

Example 6

easy

A casino game has true house win probability

0.52

. Over millions of plays, what fraction does the house win?

Example 7

easy

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

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

After 100 coin flips you have 40 heads (

40\%

). To raise the overall fraction near

50\%

, do you need future flips to favor heads?

Example 10

medium

A spinner pays $2 on win (

p=0.3

) and $0 otherwise. Its expected payout is $0.60. Over 1000 spins, roughly what total payout is expected?

Example 11

medium

Two estimates of a probability: one from 100 trials (

0.55

), one from 10000 trials (

0.505

). The true value is

0.5

. Which illustrates the LLN better?

Example 12

medium

A quality process produces

5\%

defects. In a batch of 20 you see 3 defects (

15\%

). What does LLN predict for a batch of 10000?

Example 13

medium

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

Example 14

medium

A running proportion of heads is

0.6

n=50

and

0.51

n=2000

. Does the absolute number of extra heads necessarily shrink?

Example 15

medium

A game has expected value

-\$0.10

per play. Over 5000 plays, what is the expected total result?

Example 16

medium

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

n

Example 17

medium

A slot machine has expected value

-\$0.05

per spin. Over 20000 spins, what total result is expected, and what does the LLN say about reliability?

Example 18

challenge

A fair coin shows 520 heads in 1000 flips (

0.52

). After 1000 more fair flips, what is the expected overall proportion of heads?

Example 19

challenge

A bettor reasons: 'red has missed 8 spins, so it is due.' Identify the error and the correct probability of red on a wheel where

P(\text{red})=\tfrac{18}{38}

Example 20

challenge

A die's running average is

4.0

after 9 rolls (sum 36). What single next roll would bring the 10-roll average exactly to the expected

3.5

Example 21

easy

A biased coin lands heads with true probability

0.7

. After 1,000,000 flips, what fraction of heads do we expect?

Example 22

easy

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

Example 23

easy

True or false: doubling your sample size doubles how close the sample mean is to the true mean.

Example 24

easy

A roulette bet pays out with expected value

-\$0.05

per dollar. After 10,000 $1 bets, what total loss is expected?

Example 25

easy

A random variable

X

has

E(X)=12

. After 500 independent samples, what is the expected sample mean?

Example 26

medium

A simulation rolls a fair die

n

times and tracks the running mean. After

n=10

the mean is

4.1

. After

n=10{,}000

the mean is

3.51

. Which is consistent with LLN?

Example 27

medium

A medical test has true sensitivity (probability positive given disease) of

0.92

. A clinic tests 10,000 sick patients. About how many positive results are expected?

Example 28

medium

A factory's defect rate is

2\%

. To estimate it within

\pm 0.5\%

with high confidence, would you sample

100

items or

50{,}000

items?

Example 29

medium

You simulate 10,000 rolls of a fair die. You compute the average. By LLN it should be close to

3.5

. Approximately what standard deviation does

\bar{X}_{10000}

have? (Die SD

\approx 1.708

Example 30

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 31

medium

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

6

Example 32

hard

A polling firm samples 2,500 voters and finds

52\%

support a candidate, whose true support is

50\%

. Suppose 10 such polls of size 2,500 are run. By LLN, what is the approximate average across the 10 polls?

Example 33

hard

A fair coin is flipped

n

times. Let

S_n

be the number of heads. By LLN,

S_n/n \to 0.5

. Does this mean

|S_n - n/2|

stays small as

n

grows?

Example 34

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 35

hard

A Monte Carlo estimator for

\pi

uses random points in a unit square and counts those inside the inscribed quarter circle. After

n

trials,

4 \hat p \to \pi

where

\hat p

is the empirical fraction. Why does this work?

Example 36

hard

Two random variables

X_1, X_2, \dots

have

X_i = 1

with probability

1/i

and

0

otherwise, but are independent. Are they identically distributed?

Example 37

hard

A fair die is rolled 36 times. Approximately what is the expected sum of the rolls?

Example 38

challenge

A sequence of i.i.d. random variables has

E(X) = 0

but

\text{Var}(X) = \infty

. Does the Strong Law of Large Numbers still hold?

Example 39

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 40

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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Related Concepts

Basic Probability Mean as Fair Share Central Limit Theorem Confidence Interval

Background Knowledge

These ideas may be useful before you work through the harder examples.

probability basicmean fair share