Year 7ProbabilityExperimental probabilityYear 7 Maths

Chance experiments and the Law of Large Numbers

What this lesson covers

Some chance-based situations, such as rolling a weighted die, can only be explored through probability experiments.
Results from sets of repeated trials for the same experiment may vary.
The Law of Large Numbers states that as the number of trials in a chance experiment increases, the experimental probability will approach the theoretical probability.
The estimated probability of an event from an experiment is the number of times the event happens divided by the total number of trials.

Key idea

What is the chance of a six on this die?

It has six faces, so you would say one sixth.
But this one has a weight hidden inside it.
Now no amount of looking will tell you.

Key idea

Some probabilities cannot be worked out

Counting faces only works if they are equally likely.
A weighted die breaks that assumption.
So there is nothing to count.

Situation	Can you calculate it?
A fair coin	yes, two equal outcomes
A fair die	yes, six equal outcomes
A weighted die	no
A drawing pin landing point up	no

Key idea

So you roll it. A lot.

120 rolls, and count how often each face came up.
Five faces landed between 14 and 18 times.
The six landed 40 times.

Key idea

Experimental probability

How many times it happened, divided by how many trials.
40 sixes out of 120 rolls.
40 ÷ 120 = one third.

Face	Working	Probability
1	14 / 120	0.12
2	16 / 120	0.13
3	15 / 120	0.13
4	17 / 120	0.14
5	18 / 120	0.15
6	40 / 120	0.33

Worked example

Against a fair die

A fair die gives one sixth, about 0.17.
This one gives 0.33 for a six.
Twice as often. It is definitely weighted.

Die	Working	P(six)
Fair die, any face	1 / 6	0.17
This die, a six	40 / 120	0.33
So a six comes up	about twice as often

Key idea

They still add to 1

Every roll produced exactly one face.
So the counts total 120, and the probabilities total 1.
That is true whether the die is fair or not.

Face	Working	Probability
1	14 / 120	0.12
2	16 / 120	0.13
3	15 / 120	0.13
4	17 / 120	0.14
5	18 / 120	0.15
6	40 / 120	0.33

Key idea

What a fair die would have looked like

Twenty of each, all six bars the same height.
The dashed line is where they would all sit.
One bar is a long way above it.

Key idea

Now something about fair coins

Flip a fair coin 30 times and count the heads.
Then do it again. And again.
You will get three different answers.

Run	Result	P(heads)
First 30 flips	18 heads	0.60
Second 30 flips	13 heads	0.43
Third 30 flips	16 heads	0.53

Key idea

And that is completely normal

0.60, then 0.43, then 0.53: all from a fair coin.
None of those runs was wrong.
Repeated sets of trials just do vary.

Run	Result	P(heads)
First 30 flips	18 heads	0.60
Second 30 flips	13 heads	0.43
Third 30 flips	16 heads	0.53

Key idea

So 18 heads out of 30 proves nothing

It is not evidence the coin is rigged.
30 trials is simply not many.
To find out, you need more of them.

Run	Result	P(heads)
First 30 flips	18 heads	0.60
Second 30 flips	13 heads	0.43
Third 30 flips	16 heads	0.53

Worked example

Watch what more trials do

10 flips gave 0.70. 30 gave 0.60.
100 gave 0.54. 500 gave 0.506.
Closing in on a half, and not by accident.

Trials	Heads	P(heads)
10 flips	7 heads	0.70
30 flips	18 heads	0.60
100 flips	54 heads	0.54
500 flips	253 heads	0.506

Key idea

The Law of Large Numbers

As the number of trials goes up...
...the experimental probability gets closer to the true one.
Which is why 100 trials beats 30, and 1000 beats 100.

Trials	Heads	P(heads)
10 flips	7 heads	0.70
30 flips	18 heads	0.60
100 flips	54 heads	0.54
500 flips	253 heads	0.506

Key idea

What it does not promise

You will not land exactly on 0.5. Almost never.
And a run of heads does not have to be paid back.
The coin has no memory of what it just did.

The Law	Says
It does say	more trials get you closer
It does not say	you will land exactly on it
It does not say	a run of heads must be balanced out

Key idea

Five heads in a row. What next?

People say tails is "due". It is not.
The next flip is still exactly 0.5.
The coin does not know about the last five.

Key idea

Back to the weighted die

The same warning applies there.
120 rolls is decent, but not proof.
600 rolls would make the case much stronger.

Key idea

How many is enough?

At least 30 for a rough answer.
At least 100 before you compare each outcome.
More whenever you can be bothered.

Trials	Good enough for
under 10	nothing much
30	a rough estimate
100	comparing each outcome
500+	being fairly confident

Your turn

Your turn: four questions

Pause here and try all four.
Question 3 wants a judgement and a reason.
Think about how much 30 flips varied earlier.

	Question
1	In 40 spins, red comes up 14 times. Experimental probability of red?
2	A coin is flipped 50 times and lands heads 31 times. Give P(heads).
3	Is that strong evidence the coin is unfair? Say why.
4	Flipped 5000 times, what should the experimental probability be near?

Show answer

1. 14 ÷ 40 = 0.35, or 7/20. 2. 31 ÷ 50 = 0.62. 3. Not strong evidence. Fifty trials is a small number and this much variation is ordinary: the sets of thirty in this lesson ranged from 0.43 to 0.60 on a fair coin. 4. Near 0.5. With that many trials the experimental probability should be very close to the theoretical one, though almost certainly not exactly 0.5.

Worked example

The answers

14 ÷ 40 = 0.35. And 31 ÷ 50 = 0.62.
No, 50 flips is not strong evidence; 30 gave 0.43 to 0.60.
At 5000 flips, expect very close to 0.5.

	Answer
1	0.35
2	0.62
3	no, 50 trials is too few
4	near 0.5

Summary

What to take away

Experimental probability: how often ÷ how many trials.
Some situations can only be measured, never calculated.
More trials get closer, but never promise exactness.

	Remember
Experimental probability	times it happened ÷ total trials
Some situations	can only be measured, not calculated
Repeated sets	give different answers; that is normal
More trials	get you closer to the truth
All the outcomes	still total 1

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