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Probability · Subtopic 2 of 4

Combined Events and Tree Diagrams

For two independent events, multiply ("and"). For mutually exclusive outcomes, add ("or"). A tree diagram organises both: multiply along each route, then add the routes you want. For "at least one", use $1-P(\text{none})$ .

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

List outcomes in a sample space diagram.
Use the AND (multiply) and OR (add) rules correctly.
Draw and use tree diagrams.
Use $1-P(\text{none})$ for "at least one".

Pause and think

Roll two dice. Which total is most likely: $2$ , $7$ or $12$ ?

Section 2 of 14

Core idea

Start hereTwo spins

Spin a spinner twice. The chance of red then red is smaller than red once: you need two things to happen. Tree diagrams track every path, so you can multiply along and add across.

The big idea

Independent events: multiply along branches ("and"); add end results ("or"). "At least one" $=1-P(\text{none})$ .

1. Tree diagrams

Tree diagram: two spins of a spinner with P(red) = 0.3

P(\text{one of each})=0.21+0.21=0.42

. The four ends add to

1

Worked exampleSpinner P(red) = 0.3, twice

Both red; one of each; at least one red.

Goal: Three probabilities.
Strategy: Multiply along, add across.

$0.3\times0.3=0.09$
RR.
$0.3(0.7)+0.7(0.3)=0.42$
RN or NR.
$1-0.7^2=0.51$
Complement of no reds.

Answer

0.09

0.42

0.51

✓ Check

0.09+0.42=0.51

✓.

2. Sample spaces

Sample space for two dice: totals of 8

5 of the 36 cells total 8, so

P(8)=\tfrac{5}{36}

Worked exampleTwo dice

P(total 8); P(double).

Goal: Probabilities.
Strategy: Count cells of a $6\times6$ grid.

$(2,6),(3,5),(4,4),(5,3),(6,2)\Rightarrow\tfrac5{36}$
Total 8.
$\tfrac6{36}=\tfrac16$
Diagonal.

Answer

\frac5{36}

;

\frac16

✓ Check36 equally likely cells ✓.

Common misconceptionAdding along branches Along a path, multiply; only add separate paths.

Pause and think

Coin and die: P(head and 6)?

Summary

× along, + between.
"At least one": 1 − none.

Independent: One does not affect the other.

HistoryThe letters that started it all. In 1654 the gambler Chevalier de Méré asked Blaise Pascal about splitting the stakes of an unfinished game. Pascal wrote to Pierre de Fermat, and their letters worked out how to count combined outcomes. Many historians date modern probability theory from that correspondence.

Fun factDe Méré's dice puzzle. De Méré noticed he won bets on "at least one six in

4

rolls" but lost on "at least one double six in

24

rolls of two dice". The maths agrees:

1-\left(\tfrac56\right)^4\approx0.518

but

1-\left(\tfrac{35}{36}\right)^{24}\approx0.491

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