KS3 · Maths

Probability of two combined events

Spin a spinner twice and ask for 'at least one X'. Multiply the branches, add them, or both? The twist: both, but in two different places.

Maths · Probability

Spin the X/Y spinner twice

Tap a row to light up one path and see what gets multiplied. Tap all four rows, then add the four answers. What do you notice?

Each spin: P(X) = 4/9 and P(Y) = 5/9. Picture nine equal sectors, four marked X and five marked Y. The bag drawing stands for the spinner.

1st draw2nd draw4/95/94/95/94/95/9XYXYXYXXXYYXYY4X · 5Y4X5Y

The spinner is the same on every spin, so the second stage has the same probabilities as the first.

Tap a row

Tap an outcome row — the two branches on its path glow indigo and the multiplication is laid out.

Watch out: One path gives ONE outcome. An event like 'at least one X' is bigger than a single path.

Venn diagrams

Which circle does each card sit in?

Cards numbered 1 to 10 are in a bag. Event A is the cards 1, 2, 4, 5 and 10. Event B is the cards 2, 4, 6, 8 and 10. For each card, tick the circle or circles it belongs to, or "In none" if it is in neither.

  • A Event A: cards 1, 2, 4, 5, 10
  • B Event B: cards 2, 4, 6, 8, 10
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10

Worked example: an outcome table and a two-way table

Problem

Spin a fair 1–5 spinner twice and add the two scores. Find P(the sum is even and more than five) and P(the sum is even or more than five).

Your turn to fill the gaps

Find the missing region by working backwards

A spinner has 30 equal sectors and P(A) = 18/30. A Venn diagram shows that 13 sectors are in A but not in B. How many sectors are in both A and B, and what is P(A and B)?

  1. P(A) = 18/30. The 30 counts every outcome in the sample space, so the 18 is the number of outcomes inside circle A.
  2. Circle A is two regions put together: 'A only' (13 outcomes) and 'A and B' (x outcomes).
  3. missing step
Which line is step 3?

Check the working

Which line goes wrong?

A spinner has 20 equal sectors, shown in a three-circle Venn diagram of events A, B and C. The counts are: A only 3, B only 2, C only 4, A and B but not C 2, A and C but not B 3, B and C but not A 1, all three circles 2, none of the circles 3. Find P(A and C).

A student's answer — which line goes wrong?

Table or tree?

Outcome tablevsProbability tree

Same trial, two drawings. Which would you pick?

Focus

Answer for the same trial

Outcome table

Count 10 matching outcomes out of 24: 10/24

Probability tree

Multiply along the path: 5/8 × 2/3 = 10/24

The insight

Same trial, same probability. The drawing changes the working, never the answer.

What it shows you

Outcome table

Every outcome, so you can see where the probabilities come from

Probability tree

The probability of each stage on the branches

Few distinct outcomes

Outcome table

Can be long-winded

Probability tree

More efficient

Many distinct outcomes

Outcome table

Neater

Probability tree

Not the neater choice

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Tables, Venn diagrams and trees all list the same outcomes. The skill is knowing when to count, when to multiply and when to add.

What you need to know

  • Theoretical probability = number of desired outcomes ÷ number of outcomes in the sample space, when every outcome is equally likely.
  • Give a probability as a fraction, decimal or percentage. If the decimal would recur, keep the fraction to avoid rounding errors.
  • 'And' needs both events: one cell of a two-way table, or the overlap of two Venn circles.
  • 'Or' includes outcomes in either event or both, and an outcome in both is counted once, not twice.
  • Have a goRavi spins a 1–10 spinner. "P(even) is 5/10 and P(more than 6) is 4/10," he says, "so P(even or more than 6) is 9/10." Which sectors did he count twice?

    8 and 10. The real answer is 7/10.

    8 and 10 are even and more than 6, so they sit in both events. 'Or' counts each outcome once, which gives 7 outcomes.

  • The Venn rectangle is the whole sample space, so 'not A' is everything outside circle A.
  • On a probability tree for a two-stage trial, multiply along the connected branches to get one end outcome.
  • An event can contain several outcomes. Add their probabilities, never multiply them.
  • Have a goOn your tree, XX is 16/81, XY is 20/81 and YX is 20/81. Add or multiply them to get P(at least one X)? Work it out.

    Add: 16/81 + 20/81 + 20/81 = 56/81

    'At least one X' is an event made of three outcomes, so you add their probabilities. Multiplying them gives a tiny number, smaller than any single path.

  • Probabilities in a sample space total 1, so the ends of a tree should add to 1: a handy check.
  • Have a goA tree's four end probabilities are 2/9, 2/9, 3/9 and 1/9. Quick check: do they pass?

    No. They total 8/9, not 1.

    The ends list every outcome of the trial, so they must total 1. A shortfall means a slip or a missing outcome.

  • Before multiplying, put tree probabilities in the same form; if one is 1/3, change the others to fractions.

The big picture

Tables, Venn diagrams and probability trees all list the same sample space. 'And' means the overlap, 'or' counts each outcome once, a path on a tree multiplies to give one outcome, and an event adds up the outcomes it contains.

Key points

1P(A) = desired outcomes ÷ outcomes in the sample space, for equally likely outcomes.
2'And' is the overlap; 'or' is either circle with shared outcomes counted once.
3Tree: multiply along a path for one outcome; add outcomes for an event.
4The ends of a tree total 1, and a table or a tree gives the same probability for the same trial.

Worked example

Problem

A spinner has 16 equal sectors: five show 4, two show 2, three show 3 and six show 0. Sam wins on a 2 or a 4. Alex wins on a 3 or a 4. Find P(both win), P(Sam or Alex wins), P(only Sam wins) and P(neither wins).

⚠ Watch out

Always multiplying, or adding two event counts without checking the overlap. Multiply only along the connected branches for one outcome, add the outcomes that make up an event, and count an outcome that is in both events once.

🧠

Memory hook

Multiply along, add across, count the overlap once.

✓

Check yourself

A tree has the end outcomes XX, XY, YX and YY. Which ends would you use for 'exactly one X', and why do you add them rather than multiply them?

Flashcards

(15)
What is a theoretical probability?
P(A) = number of desired outcomes ÷ number of outcomes in the sample space, when every outcome is equally likely.
Cards 1 to 10: P(square number) = 3/10. Will 10 real picks give exactly 3 squares?
Not necessarily. 3/10 comes from counting 1, 4 and 9 in the sample space; a set of trials may not match it.
The decimal for a probability would recur. What do you do?
Leave it as a fraction (4/18 = 2/9 = 0.2 recurring). That avoids a rounding error, and the fraction does not need to be simplified.
Cards 1 to 10 sorted by odd/even and square/not square: where is "odd and square"?
In the cell where the odd row meets the square column. It holds 1 and 9, so P(odd and square) = 2/10.
On a Venn diagram, where is 'A and B'? Where is 'not A and not B'?
'A and B' is the overlap of the two circles. 'Not A and not B' is the region outside both circles.
What does 'or' include, and how do you count shared outcomes?
Outcomes in either event or in both. Count each outcome once, not twice.
What does the rectangle round a Venn diagram show? What is the complement of A?
The rectangle encloses the sample space, all the possible outcomes. The complement of A is every outcome outside A.
In a three-circle Venn diagram, which region counts for 'A and C'?
Only the region inside both circle A and circle C, whether or not those outcomes are also in B.
A Venn region holds 7 outcomes in a sample space of 20. What is its probability?
7/20. The numerator is the number of outcomes in the region and the denominator is the whole sample space.
What does a branch on a probability tree show? What does the end of a two-layer tree show?
A branch shows a possible outcome of one stage and its probability. The end of the tree is the sample space for the whole trial.
Spinner with P(A) = 5/8, then a coin with P(heads) = 1/2. What is P(A and heads)?
Multiply along the connected branches: 5/8 × 1/2 = 5/16.
How do you find the probability of an event made of several outcomes?
Add the probabilities of its outcomes. Never multiply them.
Why do you add up the end probabilities of a tree?
They must total 1 (for example 5/16 + 5/16 + 3/16 + 3/16 = 16/16), because the ends list every outcome. It checks your working.
A tree has branches 1/3 and 32%. How do you get them into the same form?
1/3 loses precision as a decimal, so change 32% to a fraction instead: 32/100 = 8/25.
Which suits a trial with only a few distinct outcomes: an outcome table or a probability tree?
A tree is more efficient. A table shows where probabilities come from but can be long-winded; it is neater when there are many distinct outcomes.

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.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More KS3 Maths topics

See the full KS3 Maths curriculum →

How this lesson was checked. This KS3 Mathslesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 9 October 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.