Tree diagrams and sequential events

Using a tree diagram for two-stage experiments: multiply probabilities ALONG the branches of a path (AND), add across the paths that make up an event (OR), and use 1 − P(none) for "at least one". The crucial distinction between sampling WITH replacement (probabilities stay the same) and WITHOUT replacement (the totals, and so the later probabilities, change).

Statistics · Probability

Tree diagrams and sequential events

Why it works

A tree diagram lays out a two- (or more) stage experiment stage by stage. From each point, the branches show the possible outcomes of the next stage with their probabilities, and the branches leaving any one point always sum to $1$ .

Two rules do all the work:

Multiply along the branches of a single path — this gives the probability of

that whole sequence happening (a string of "and"s).

Add the paths that make up your event — different routes to the same overall

outcome are alternatives ("or"), and the separate paths are mutually exclusive, so you add.

"At least one". Listing every path with "at least one" of something is slow; it is nearly always faster to use the complement:

P(\text{at least one}) = 1 - P(\text{none}).

With vs without replacement — the key distinction. Suppose a bag has

5

red and

3

blue counters (

8

in total) and you take two.

With replacement — you put the first counter back before the second draw, so the

bag is unchanged:

P(\text{red})

\tfrac58

on both draws. The draws are independent.

P(\text{both red}) = \tfrac58 \times \tfrac58 = \tfrac{25}{64}

Without replacement — you keep the first counter, so the second draw is from only

7

counters and the count of what you drew has gone down. If the first was red, only

4

reds remain out of

7

P(\text{both red}) = \tfrac58 \times \tfrac47 = \tfrac{20}{56} = \tfrac{5}{14}

.

The whole skill in "without replacement" problems is updating the numerator and the denominator on the second branch to match what was removed.