Conditional probability and tree diagrams

Using tree diagrams where the second-stage branches are conditional probabilities — multiplying along branches for P(A ∩ B), adding the paths to an event, and reversing the condition to find P(first | second).

Statistics · Probability

Conditional probability and tree diagrams

Why it works

A tree diagram is really a picture of the multiplication formula. The first set of branches shows the first event; the second set, growing from each, shows the next event given the first — so the second-stage probabilities are conditional. That is why you multiply along a path:

P(A \cap B) = P(A)\,P(B \mid A)

is exactly the two probabilities on the branches multiplied together.

This matters most when the stages are dependent — above all "without replacement", where taking the first item changes the probabilities for the second. Drawing two counters from a bag of

3

red and

2

blue: the first branch is

\tfrac35

red, but the second branch given red is

\tfrac24

— only

4

counters are left and one red is gone. The conditional probabilities live on the second branches.

To find the probability of an event that can happen several ways, add the paths that lead to it:

P(\text{one of each colour}) = P(RB) + P(BR)

.

Reversing the condition. The subtle Year 2 skill is going backwards: you're told the second outcome and asked for the first —

P(\text{first} \mid \text{second})

. You can't read this off a branch directly, because the branches give "second given first". Instead use the definition:

P(A \mid B) = \frac{P(A \cap B)}{P(B)},

where

P(A \cap B)

is the single path through both, and

P(B)

is the sum of all paths that reach

B

. This is how "given the test was positive, what's the probability the person is actually ill?" type questions work — and the answer is often surprising.