KS3 · Maths

Calculating theoretical probabilities

Two letters on a spinner, A and B. So landing on A is a 50–50 shot, right? Hold that thought. It's the trap this whole lesson is built to catch.

Maths · Probability

Six equal sectors, two letters

A spinner has six equal sectors. Five say A and one says B. Is landing on A a 50–50 shot? Do not answer yet. Sort each sector first, then read the counts.

Still to sort

Lands on A (0)

These sectors are in the event 'lands on A'.

Where the line is: Only sectors with an A on them. A sector with a B is not in this event.

Lands on B (not A) (0)

The opposite event to landing on A.

Where the line is: The sectors left over once the A sectors are taken out.

6 of 6 still to sort.

Five sectors say A and one says B. All six are the same size.

Watch out: Two letters on a spinner does not mean two equally likely outcomes. The equally likely outcomes are the six sectors.

Same spinner, two trees

A branch for every sectorvsLike branches collected

On a tree, the outcome goes at the end of each branch and its probability goes on the branch. Every outcome has to be shown.

Focus

Branches drawn

A branch for every sector

6, one for each equal sector

Like branches collected

2, one for A and one for B

The insight

Both trees show every outcome of the spin. Only the way the outcomes are grouped changes.

Probability on each branch

A branch for every sector

1/6 on every branch

Like branches collected

A: 5/6 and B: 1/6

How you get P(A)

A branch for every sector

Add the five A branches: 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Like branches collected

Read it straight off the A branch: 5/6

An event and its opposite

A branch for every sector

A has five branches, B has one

Like branches collected

A: 5/6, not A: 1/6

Maths · Probability

Can you find it by counting?

You can only find a theoretical probability when the individual outcomes are equally likely. Pick a trial, then decide which side it belongs on.

Still to sort

Theoretical probability can be found (0)

The individual outcomes are equally likely.

Where the line is: If you can say the individual outcomes are equally likely, you can count them.

Cannot be found (0)

The individual outcomes are not (or may not be) equally likely.

Where the line is: If the individual outcomes are not equally likely, counting outcomes does not give the probability.

5 of 5 still to sort.

The test is always the same: are the individual outcomes equally likely?

Maths · Probability

Find the wrong line

A box holds 80 sweets in 4 flavours: fudge 12, orange 8, toffee 25 and mint 35. One sweet is picked at random. Find P(fudge). Here is Jess's working. Which line loses the mark?

Jess's working — which line goes wrong?

Maths · Probability

Read a probability off a Venn diagram

The numbers 1 to 12 are equally likely outcomes. A is "multiple of 3" and B is "even". Tick the sets each number belongs to, then check.

  • A A: multiples of 3
  • B B: even numbers
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12

Maths · Probability

Every probability lives between 0 and 1

Drag each probability to where it sits from 0 to 1. P(7) and P(not 7) are for a fair six-sided dice. The others can come from anywhere. Then check.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Count the equally likely outcomes, not the labels.

What you need to know

  • A theoretical probability counts the outcomes you want in a sample space where every individual outcome is equally likely.
  • Write it as a fraction: outcomes in the event on top, total outcomes underneath. A fair dice gives P(5) = 1/6.
  • Have a goA spinner has five equal sectors labelled A to E. What is P(C)? Write it as a fraction.

    1/5

    One sector is C out of five equal sectors, so the event goes on top and the total underneath. Putting 5 on top flips the fraction upside down.

  • When labels repeat, count every equal sector. Three A sectors and two B sectors give P(A) = 3/5 and P(B) = 2/5.
  • Have a goSam says: "Two letters, X and Y, so P(X) = 1/2. Easy." But the spinner has 4 equal sectors: three say X and one says Y. Fix Sam's answer.

    P(X) = 3/4

    Sam counted letters. The equally likely outcomes are the four sectors, and three of them say X.

  • You can find a theoretical probability only if individual outcomes are equally likely. Bigger green cubes than blue? Then you can't.
  • For a set of outcomes, count them all: on a fair dice, P(even) = 3/6 and P(not 4) = 5/6.
  • Have a goA fair dice is rolled. What is P(a number greater than 4)?

    2/6

    The numbers greater than 4 are 5 and 6, which is two outcomes out of six. Counting only one of them gives 1/6 and misses part of the set.

  • Impossible events have probability 0 and certain events have probability 1. On a fair dice, P(7) = 0, P(not 7) = 1.
  • Fractions, decimals and percentages all work: 3/5 = 0.6 = 60%.
  • Whatever the form, a probability lies from 0 to 1, so a count of outcomes such as 3 isn't one.
  • Expected number = probability × trials. With P(A) = 3/5, expect 300 As in 500 spins, but real results may differ by chance.
  • Have a goA fair dice is rolled 120 times. How many 4s do you expect?

    20

    P(4) = 1/6 and 1/6 of 120 is 20. That is only the expected number, so a real run of 120 rolls may give a different count.

  • In frequency and outcome tables, it's the event's frequency or cells over the total, never over the number of categories.

The big picture

A theoretical probability counts the outcomes in your event over the total number of outcomes, but only when every individual outcome is equally likely. Count equal sectors, faces and cards rather than labels, then read the same count from a list, table, tree or Venn diagram.

Key points

1Theoretical probability = outcomes in the event ÷ total outcomes, when every individual outcome is equally likely.
2Count the individual equally likely outcomes (sectors, faces, cards, sweets), not the labels or categories.
3If the individual outcomes are not equally likely, the theoretical probability cannot be found.
4Probabilities lie from 0 (impossible) to 1 (certain) and can be fractions, decimals or percentages.
5A probability is the proportion of times an event is expected to happen in a suitably large experiment. Expected number = probability × trials, but actual results may differ due to chance.
6To compare probabilities, write them in the same form: a bigger number of outcomes does not mean a bigger probability when the sample spaces differ in size.

Worked example

Problem

The cards 1 to 10 are shuffled and one is picked at random, so each card is equally likely. The cards are sorted into an outcome table by prime or not prime and odd or even. Find P(odd prime).

⚠ Watch out

Dividing by the number of labels instead of the number of equally likely outcomes. Two letters on a spinner doesn't make P(A) = 1/2, and 12 fudge sweets are not '12 out of the number of flavours'. Count the sectors, or use the total frequency.

🧠

Memory hook

Count the sectors, not the letters.

✓

Check yourself

Cover the page. A spinner has 10 equal sectors and 4 say Z. Explain why P(Z) = 4/10, not 1/2. What would stop you finding it at all?

Flashcards

(14)
What must be true before you can find a theoretical probability?
The individual outcomes must all be equally likely.
How is a theoretical probability written as a fraction?
Number of outcomes in the event over the total number of outcomes.
A spinner has 8 equal sectors and 2 are red. What is P(red)?
2/8 (= 1/4). Count the sectors, not the colours.
The usual slip is dividing by the number of labels. What do you divide by instead?
The total number of equally likely individual outcomes, or the total frequency in a table.
Why is P(pizza) not necessarily 1/4 for a menu of four dishes?
One dish may be more popular, so the outcomes may not be equally likely.
What probabilities do impossible and certain events have?
0 for impossible and 1 for certain. Every probability lies between them.
Write 12/80 as a decimal and a percentage.
0.15 and 15%, because 12/80 = 3/20.
How do you find the expected number of times an outcome occurs?
Probability × number of trials. Actual results may differ due to chance.
In a frequency table, what goes underneath the fraction?
The total frequency, not the number of categories.
On a Venn diagram, what does P(not A) count?
Every outcome outside circle A.
What does P(only B) count on a Venn diagram?
Just the part of circle B outside A, with the overlap left out.
How do you compare 35% and 5/12?
Write both in the same form: 21/60 and 25/60, so 5/12 is greater.
A tree for a six-sector spinner with five A sectors: why is the A branch 5/6?
Five branches of 1/6 are collected into one A branch by adding: 1/6 added five times is 5/6.
The outcomes are 1 to 12 and P(A) = 1/4. How many outcomes are in A?
1/4 = 3/12, so there are 3 outcomes in A.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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