Tree Diagrams

Grade 4

When to use a tree diagram

Tree diagrams are perfect when events happen one after another — you spin a spinner then flip a coin, or you pick two counters from a bag. Each branch shows one possible outcome and carries its probability.

Two rules govern every tree diagram:

Multiply along the branches to find the probability of a particular path (both events happen in that order).

Add the paths that give the outcome you want.

Probabilities on each set of branches must add to 1

At any point in the tree, the branches coming from the same node must add up to 1. If you've written 0.3 on one branch, the other must be 0.7. Use this to fill in missing values.

Worked example — two coins

EXAMPLE
A fair coin is flipped twice. Draw a tree diagram and find (a) P(two heads), (b) P(exactly one head).
First flip: H (prob ½) or T (prob ½). From each, a second flip: H (½) or T (½). Four paths: HH, HT, TH, TT. (a) P(HH) = ½ × ½ = 1/4. (b) One head means HT or TH. P(HT) = ½ × ½ = ¼. P(TH) = ½ × ½ = ¼. Add: ¼ + ¼ = 1/2.

Worked example — unequal probabilities

EXAMPLE
A bag has 4 red and 6 blue counters. One is picked at random, replaced, then a second is picked. Find P(one of each colour).
P(red) = 4/10 = 2/5. P(blue) = 6/10 = 3/5. (Replacement means same probabilities second time.) "One of each" means RB or BR. P(RB) = 2/5 × 3/5 = 6/25. P(BR) = 3/5 × 2/5 = 6/25. P(one of each) = 6/25 + 6/25 = 12/25.
Nova
Try this one

A bag contains 3 red counters and 7 blue counters. A counter is taken out at random, its colour is recorded, and it is put back in the bag. A second counter is then taken out at random.

(a) Complete the tree diagram below by filling in the four missing probabilities.

First pick           Second pick

Red (\( \frac{3}{10} \)) ——— Red ( __ )
Red ( __ ) <
Red (\( \frac{3}{10} \)) ——— Blue ( __ )

Blue ( __ ) <
Blue (\( \frac{7}{10} \)) ——— Red ( __ )
Blue ( __ ) <
Blue (\( \frac{7}{10} \)) ——— Blue ( __ )

[2]

(b) Work out the probability that both counters are the same colour. [3]

Nova's hint:

This uses tree diagrams with replacement — because the counter goes back, each pick is independent and the probabilities stay the same on every branch.

Ask yourself: 'How do I combine the probabilities along a branch to find the probability of two events both happening?'

For example, if P(A) = 1/4 and P(B) = 1/4 along a branch, the probability of that route is 1/4 × 1/4.

Try to Solve it with Nova?