Law of large numbers

What is it about?

If you throw a die $n$ times and get a six $k$ times, then

$k$ is the absolute frequency and
$h_n = \dfrac{k}{n}$ is the relative frequency of the six.

Example: $17$ sixes in $100$ throws give $h_{100} = \dfrac{17}{100} = 0.17 = 17\,\%$ .

With few throws $h_n$ varies a lot. The more often you throw, the more the relative frequency settles – close to the probability $p$ (coin: $p = 0.5$ , die: $p = \frac{1}{6} \approx 0.167$ ). This is called the empirical law of large numbers.

There is no guarantee, though: usually $h_n$ gets closer to $p$ with more throws, but not always. Even after many throws $h_n$ can move a little further away again – large deviations just become less and less likely.

Conversely, this is used to estimate unknown probabilities: for a drawing pin you cannot calculate $p$ for “point up” – but after very many throws $h_n$ is a good estimate of $p$ .

Important: Chance does not “even out”. A die has no memory – after five throws without a six, a six is not more likely on the next throw. The relative frequency approaches $p$ because early outliers are spread over more and more throws.

The funnel

The green funnel shows the range $p \pm 1.96 \cdot \sqrt{\dfrac{p(1-p)}{n}}$ . After $n$ throws, the relative frequency lies inside it in about 95 % of all runs. It gets narrower as $n$ grows – four times as many throws make it half as wide.

Try it

Throw the die one at a time and watch how much the curve jumps at first. Why is that?
Throw 1000 times or press Play. Where does the curve settle?
Compare with the coin: does the curve become “calm” faster? Coin
Estimate the drawing pin’s probability without showing it first. Drawing pin
Switch off the logarithmic axis. What can you see better, what worse? Linear axis

Exercises

Exercise 1: Expectation

About how many sixes do you expect when you throw a die $600$ times? Is it suspicious if there are only $92$ sixes?

Show solution

You expect about $600 \cdot \frac{1}{6} = 100$ sixes. $92$ sixes are not suspicious: that is a relative frequency of about $0.153$ . At $n = 600$ the funnel ranges from about $0.137$ to $0.196$ – so $0.153$ lies within the usual range of variation.

Exercise 2: Estimating the drawing pin

Throw the drawing pin 1000 times without showing the probability. Give an estimate for $p$ (“point up”) and then compare it with the set value (switch on “Show probability p” for this). Load exercise

Show solution

The estimate is the relative frequency after 1000 throws. The set value is $p = 0.6$ ; after 1000 throws $h$ usually lies between about $0.57$ and $0.63$ .

Exercise 3: Does the die have a memory?

Lena says: “I haven’t thrown a six five times in a row, so one must come soon!” Comment on this.

Show solution

Lena is wrong (this is the so-called gambler’s fallacy). The throws are independent – the probability of a six is $\frac{1}{6}$ on every throw, no matter what happened before. The relative frequency approaches $p$ not because things even out, but because earlier deviations hardly matter when $n$ is large.

Exercise 4: Absolute and relative deviation

With many throws, watch the deviation of the count $|k - n \cdot p|$ and the deviation of the relative frequency $|h_n - p|$ . Both are shown under Results. What do you notice?

Show solution

The deviation $|h_n - p|$ typically keeps getting smaller. The deviation of the count $|k - n \cdot p|$ , however, typically even grows (roughly like $\sqrt{n}$ ). So the law of large numbers does not say that the counts “even out”.

Ideas for the classroom

Use: About grades 6 to 8 (Laplace experiments: relative frequency and the empirical law of large numbers). The drawing pin is the classic example of a probability that can only be determined by experiment.
Images: Die, coin, drawing pin and table are shown as photos; if an image is missing, the simulation draws a replacement graphic.
Common stumbling blocks:
- Gambler’s fallacy (“now the six is due”).
- “600 throws give exactly 100 sixes.”
- The probability is equated with the relative frequency of a short series of trials.
Funnel: The limits are based on the normal approximation and are only rough for small $n$ – in lower secondary school the interpretation “almost all runs lie inside” is enough.