Tree Diagrams
Grade 4When 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.
