Grade by Grade
A. Counting and Probability · Probability

Experimental Probability and the Law of Large Numbers

A1.4About 65 minutes
Learning goals
  • I can calculate an experimental probability (relative frequency) from data.
  • I can compare experimental and theoretical probabilities and explain why they usually differ.
  • I can show, with data and a graph, that experimental probability tends to approach theoretical probability as the number of trials grows.
  • I can explain why the law of large numbers does not mean that results "even out" in the short run.

Words to know

frequency
The number of times an outcome or event occurs in a set of trials.
relative frequency
Frequency divided by the number of trials; this is the experimental probability.
experimental probability
P(A) ≈ (number of trials in which A occurred) ÷ (total number of trials).
law of large numbers
As the number of independent trials grows, the experimental probability of an event tends to get closer to its theoretical probability.
random variation
The natural, unpredictable differences between results of repeated experiments.

Probability from data

When you cannot reason out a theoretical probability, you collect data. The experimental probability (also called the relative frequency or empirical probability) of event A is

P(A) ≈ (number of times A occurred) / (number of trials).

We use ≈ because it is an estimate of the true probability. A basketball player's free-throw percentage, a factory's defect rate and the share of drivers who wear seatbelts are all experimental probabilities.

Even when a theoretical probability exists, experiments are useful: they test whether the model is right. If a die gives 1s in 40% of 1000 rolls, it is probably not fair.

Remember from Grade 7: you compared experimental and theoretical probability and saw that more trials usually give results closer to the theoretical value. This lesson makes that idea precise.

Worked example

Example 1: Calculating relative frequency

A student rolled a die 60 times. The frequencies of the faces 1 to 6 were 10, 4, 12, 10, 13 and 11. Find the experimental probability of each face and of rolling an even number. Compare with the theoretical values.

  1. Check: 10 + 4 + 12 + 10 + 13 + 11 = 60 trials. ✓
  2. Relative frequencies: P(1) ≈ 10/60 ≈ 0.167, P(2) ≈ 4/60 ≈ 0.067, P(3) ≈ 12/60 = 0.2, P(4) ≈ 10/60 ≈ 0.167, P(5) ≈ 13/60 ≈ 0.217, P(6) ≈ 11/60 ≈ 0.183.
  3. Even: 4 + 10 + 11 = 25. P(even) ≈ 25/60 ≈ 0.417.
  4. Theoretical: each face 1/6 ≈ 0.167, and P(even) = 0.5.

The experimental values range from about 0.067 to 0.217 instead of all being 0.167, and P(even) ≈ 0.417 instead of 0.5. With only 60 rolls, differences this size are normal random variation; they are not evidence that the die is unfair.

05101510142123104135116expected 10face rolled (60 rolls)
Results of 60 rolls of a fair die. Each face is expected about 10 times (60 × 1/6), but the actual counts vary from 4 to 13. That is ordinary random variation in a small experiment.

The law of large numbers

The law of large numbers says: as the number of trials increases, the experimental probability of an event tends to get closer to its theoretical probability.

The graph below shows one computer simulation of 5000 die rolls. After each block of rolls, it plots the experimental probability of rolling a 2 so far. The theoretical probability is 1/6 ≈ 0.167.

  • After 10 rolls: 1 two, so 0.100.
  • After 100 rolls: 8 twos, so 0.080 (still far off!).
  • After 1000 rolls: 155 twos, so 0.155.
  • After 5000 rolls: 791 twos, so about 0.158.

Early on, the value jumps around a lot. Later, each new roll changes the fraction only a little, and the value settles close to 1/6. (It may still not equal 1/6 exactly; it just gets close.)

0.000.050.100.150.200.250.30theoretical 1/6 ≈ 0.167102050100200500100020005000number of rolls (not to scale)experimental P(2)
One simulation of 5000 die rolls: the experimental probability of rolling a 2 wanders a lot for small numbers of rolls and settles near the theoretical value 1/6 as the number of rolls grows. The horizontal spacing is not to scale.
Worked example

Example 2: Reading the trend

Use the simulation results. (a) After 500 rolls the experimental probability was 66/500. How far was it from 1/6? (b) After 5000 rolls, how many twos would be expected, and how far off was the count of 791? (c) Which is closer to 1/6: the result after 500 rolls or after 5000?

  1. (a) 66/500 = 0.132. Difference: 0.1667 − 0.132 ≈ 0.035.
  2. (b) Expected twos = 5000 × 1/6 ≈ 833.3. 833.3 − 791 ≈ 42 fewer than expected.
  3. (c) After 5000 rolls: 791/5000 = 0.1582, a difference of about 0.0085. That is much closer than 0.035.

The proportion got much closer to 1/6 (0.035 off, then about 0.008 off), even though the count at 5000 rolls was about 42 away from the expected count. The law of large numbers is about proportions, not counts.

Things do not "even out"

The law of large numbers does not say that a run of bad luck will be made up by a run of good luck. A coin that has landed tails 10 times in a row is not more likely to land heads next. The proportion settles because the early run gets diluted by thousands of new tosses, not because the coin corrects itself.

Worked example

Example 3: Is this die fair?

Kai rolls a die 1200 times and gets 312 sixes. (a) Find the experimental probability of rolling a 6. (b) How many sixes would a fair die give, on average? (c) Is this good evidence that the die is not fair?

  1. (a) 312/1200 = 0.26.
  2. (b) 1200 × 1/6 = 200 sixes.
  3. (c) 312 is 112 above 200, and 0.26 is far from 0.167 after a large number of trials. In 1200 rolls of a fair die, results this far from 200 almost never happen.

(a) 0.26, (b) 200, (c) Yes, with this many trials the result is strong evidence that the die favours 6. (The statistics strand of this course, in units such as One-Variable Data Analysis, will help you decide how far is "too far".)

Just for fun

Mathematicians have tested the law of large numbers by hand. Around 1900, the statistician Karl Pearson tossed a coin 24 000 times and got 12 012 heads, a relative frequency of 0.5005. That is a lot of thumb exercise for 0.0005.

Hands-on activity

Family dice marathon

You need: one die per person, paper, a pencil and a calculator. (One person can do this alone; it just takes longer.)

  1. Each person rolls their die 30 times and records how many times they roll a 2.
  2. Each person calculates their experimental probability of rolling a 2.
  3. Pool the results: add up all the 2s and all the rolls. Calculate the pooled experimental probability.
  4. Make a table: trials so far (30, 60, 90, …) and pooled probability after each person's results are added. Graph it like the graph in this lesson, with a dashed line at 1/6.
  5. Describe the trend. Whose individual result was farthest from 1/6? Was the pooled result closer?

What to look for: individual results from 30 rolls can easily range from 0.03 to 0.30. The pooled result from more rolls is usually, though not always, closer to 0.167. That is the law of large numbers in action. (A free online dice simulator or a spreadsheet can extend this to thousands of rolls; see lesson 6.)

Your turn

1

A thumbtack is dropped 250 times and lands point up 145 times. Find the experimental probability of point up.

2

Using the thumbtack result, predict how many times it lands point up in 1000 drops.

3

A coin is tossed 20 times and lands heads 13 times. Find the experimental probability of heads and compare with the theoretical value.

4

State the law of large numbers in your own words.

5

A spinner has 4 equal sections. In 400 spins, section A came up 92 times. Find the experimental and theoretical probability of A and the difference.

6

Why is 500 trials better than 50 for estimating a probability?

More practice: warm-up

1

An event happened 18 times in 72 trials. Experimental probability?

2

A die is rolled 300 times. About how many 5s do you expect?

3

A player made 42 of 60 shots. Experimental P(make)?

4

Write "relative frequency" as a formula.

5

True or false: after many trials, the experimental probability always equals the theoretical probability exactly.

More practice: on your own

1

A survey of 1250 Ontario drivers found 1100 had winter tires. Estimate P(a driver has winter tires) and predict the number in a group of 400 drivers. (Example numbers.)

2

Two coins were tossed 200 times: 2 heads 46 times, 1 head 108 times, 0 heads 46 times. Compare each experimental probability with its theoretical value.

3

Mira says, "I've rolled 30 times and only got one 6, so sixes are due." Explain what is wrong, using the law of large numbers.

4

A game's designer says a card is drawn with probability 0.05. In 2000 openings of virtual card packs, players found the card 64 times. Does this support the claim?

5

After 10 tosses a coin showed 7 heads (0.7). After 1000 tosses it showed 515 heads. Which result is closer to 0.5 as a proportion, and which is farther in raw count from the expected number?

Challenge

1

After 40 tosses, a coin has 28 heads (0.7). Suppose the next 960 tosses come out exactly half heads. What is the experimental probability of heads after 1000 tosses? What does this show?

2

A biased coin lands heads with probability 0.6. Over 50 tosses, about how many heads do you expect? Explain why getting exactly 30 heads is not very likely, even though 30 is the expected number.

3

Two classes test whether a die is fair. Class A rolls 60 times and gets 15 sixes. Class B rolls 6000 times and gets 1150 sixes. Which result is stronger evidence that the die is unfair? Explain.

Check your understanding

Quiz

1.

A spinner landed on red 34 times in 80 spins. What is the experimental probability of red?

2.

Which statement best describes the law of large numbers?

3.

A fair die is rolled 600 times. About how many times should a 4 appear?

4.

After 20 rolls, a die has given no 3s. What is the probability that the next roll is a 3?

5.

A coin was tossed 10 000 times with 5080 heads. Which statement is correct?

6.

Which estimate of P(a thumbtack lands point up) is most reliable?

For parents and teachers

Big idea (A1.4): experimental probability is the relative frequency from data. As the number of trials grows, it tends to get closer to the theoretical probability (the law of large numbers). Short runs can be very uneven.

How to help: The family dice marathon is the best way to see this. Pool everyone's rolls and watch the overall fraction settle near 1/6. Point out that individual results still vary a lot.

Common difficulties:

  • The gambler's fallacy: thinking outcomes are "due" after a run.
  • Expecting the count (not the proportion) to get closer to the expected number.
  • Treating a small experiment's result as proof that a die or coin is unfair.
  • Forgetting that experimental probability is an estimate (use ≈).