Mathematics · Combined-event probability

Tree diagrams with and without replacement

You draw the branches, but the second-level fractions come out wrong and the total is not one.

A tree diagram shows each stage of a combined event as a set of branches. Multiply along a path to find the chance of that sequence, and add paths when the question allows more than one.

This lesson belongs to the combined-event probability section. It uses the ideas from distinguishing independent and dependent events.

How do I build the tree?

Draw one set of branches for each draw, and write the probability on every branch. The probabilities leaving any one point must add to one.

For a draw without replacement, update the second-level fractions to match what is left after the first draw.

Worked example: a box of tickets

A box holds 5 tickets, of which 2 are winning (W) and 3 are not (L). Two tickets are drawn without replacement.

Path	Working	Probability
W then W	2/5 × 1/4	2/20 = 1/10
W then L	2/5 × 3/4	6/20 = 3/10
L then W	3/5 × 2/4	6/20 = 3/10
L then L	3/5 × 2/4	6/20 = 3/10

Check: 1/10 + 3/10 + 3/10 + 3/10 = 10/10 = 1, so the tree is consistent.

P(exactly one winning) = P(W then L) + P(L then W) = 3/10 + 3/10 = 3/5.

With replacement, the second-level fractions stay 2/5 and 3/5. Then P(exactly one winning) = 2 × (2/5 × 3/5) = 12/25.

The mistake that keeps the same denominators

The usual slip is to put 2/5 and 3/5 on every second-level branch even though the tickets are not returned. The tree looks neat and its paths still add to 1, but it describes drawing with replacement.

The check is the denominator. With no replacement, the second-level fractions must be out of 4, because one ticket has gone.

Check yourself

A bag holds 4 green and 2 yellow apples. Two apples are taken without replacement. Find the probability that at least one is green.

Answer

Use the complement. P(no green) means both yellow: 2/6 × 1/5 = 2/30 = 1/15.

P(at least one green) = 1 − 1/15 = 14/15.

What to study next

The last skill in this section is combining events that can or cannot overlap, in combining mutually exclusive and non-exclusive events. You can also build trees of your own with the probability tree builder.

For a teacher to check your tree diagrams on your own questions, see online one-to-one Mathematics tuition.

Tools for this page

Common questions

How do I read a tree diagram?

Multiply along a path to get the probability of that whole sequence. Add the probabilities of separate paths when the question allows either path. The probabilities on the end of all paths must add to one.

What changes on the branches when there is no replacement?

The second-level fractions use a smaller total, and the favourable count drops if the first draw was a favourable item. The first-level branches stay the same.

What is the shortcut for at least one?

Find the probability of none and subtract it from one. For two draws, at least one green is 1 minus P(no green). This needs one multiplication, not several.

Do I always draw the tree?

For two draws, a quick tree prevents mistakes and shows which paths to add. Longer sequences may need a listing or complement instead, but the same multiply and add rules apply.

Keep going

When your branches look right but the totals never reach one, a one-to-one Mathematics teacher can check each second-level fraction with you and fix the one that slipped.

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