Probability and Tree Diagrams

Basics

Probability runs from 0 (impossible) to 1 (certain).

            number of favourable outcomes
P(event) = ───────────────────────────────
              total number of outcomes

• All outcomes sum to 1, so P(not A) = 1 − P(A).

AND / OR rules

• Independent events ("AND") → multiply:

Two coins both heads = ½ × ½ = ¼.

• Mutually exclusive ("OR") → add:

P(red or blue) = P(red) + P(blue).

Tree diagrams

• Each branch shows an outcome and its probability; branches from a point sum to 1.
• Multiply along the branches; add the relevant end results.

Example — two counters drawn without replacement from 3 red, 2 blue:

P(both red) = 3/5 × 2/4 = 6/20 = 3/10

(Note the second fraction changes — one red has been removed.)

Expected frequency

expected number = probability × number of trials.

P(six) = 1/6, rolled 60 times → expected 10 sixes.

Venn diagrams & set notation

• A ∩ B = in both (intersection)
• A ∪ B = in either (union)
• A' = not in A (complement)

Exam tip

AND → ×, OR → +. Check "with or without replacement" — without replacement, the totals change on the second pick.