Probability Tree Diagrams

La méthode, étape par étape

1. 1

   Draw a branch for every outcome

   first event: red (0.6) or blue (0.4)

   Every possible outcome gets a branch, and the probabilities on any set of branches from one point must add to 1. Checking that sum immediately catches a missing outcome before it corrupts the rest of the diagram.

2. 2

   Extend the tree for the second event

   from each first branch, a full set of second branches

   Every first-stage outcome leads to a complete set of second-stage outcomes, so two events with two outcomes each give four paths. The structure is the same regardless of whether the probabilities change between stages.

3. 3

   Multiply along a path

   P(red then blue) = 0.6 × 0.4 = 0.24

   Multiplying gives the probability of both things happening in that order. This works because the second probability is conditional on the first having occurred, which is exactly what the branch structure encodes.

4. 4

   Add the paths that satisfy the condition

   P(one of each) = (0.6 × 0.4) + (0.4 × 0.6) = 0.48

   'One of each' can happen two ways, so both paths count. Missing the second path is the most common error, and it comes from reading the question as if order were specified when it is not.

5. 5

   Adjust for selection without replacement

   5 red of 8 → first 5/8, second (if red taken) 4/7

   When an item is not replaced, both the numerator and the denominator change for the second draw. Changing only the numerator is the standard slip, and it leaves the second set of branches not totalling 1.

The two operations

Multiply along a path because both events must happen; add between paths because any one of them will do. That single sentence covers nearly every tree diagram question, and students who can state it rarely apply the wrong operation.

The check is that the probabilities of all complete paths must total 1, since one of them has to happen. This catches both arithmetic errors and missing branches in one addition.

• Branches from one point total 1
• Multiply along a path (both happen)
• Add between paths (either will do)
• All complete paths total 1
• Without replacement → both numbers change

With and without replacement

With replacement, the item goes back and the second-stage probabilities are identical to the first. Without replacement, one fewer item remains, so the denominator drops by one and the numerator drops too if the item removed was of that type.

The tell is in the wording: 'and then takes another without putting the first back' is explicit, but 'takes two counters' also means without replacement. Missing that phrase makes every subsequent probability wrong.

'At least one' is easier backwards

P(at least one red) usually requires adding several paths. The complement is quicker: P(at least one) = 1 − P(none), and 'none' is a single path. For three events this turns seven additions into one subtraction.

Recognising when to use the complement is a genuine exam skill. Any question with 'at least' in it is worth checking against this approach before starting the longer route.

How we teach tree diagrams

We require every branch to be labelled with its probability and every set to be checked against 1 before any multiplication happens. Almost all errors are visible at that point and invisible afterwards.

We drill the 'at least one' complement separately, because students who have not met it will spend three minutes on a question that takes twenty seconds, and time is what these papers are short of.