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Tree Diagrams

A tree diagram lays out every possible sequence of outcomes, making multi-step probability calculations easier to track.

Mapping Out Every Path

Example 1 — exactly one head in two flips

A fair coin is flipped twice. Find P(exactly one head). 1 There are two ways to get exactly one head: Heads-then-Tails, or Tails-then-Heads 2 P(Heads then Tails) = 0.5 \times 0.5 = 0.25 3 P(Tails then Heads) = 0.5 \times 0.5 = 0.25 4 Add the two matching paths: 0.25 + 0.25 Answer: 0.5

Example 2 — at least one red from two spins

A spinner has P(red) = 0.4 and P(not red) = 0.6 . It is spun twice, independently. Find P(at least one red). 1 It is easier to find the complement first: P(no red at all) = P(not red) \times P(not red) = 0.6 \times 0.6 = 0.36 2 P(at least one red) = 1 - P(no red at all) = 1 - 0.36 Answer: 0.64

Practice

A biased coin has P(heads) = 0.7 . It is flipped twice. Find P(exactly one head).

Your answer

Mapping Out Every Path

Example 1 — exactly one head in two flips

A fair coin is flipped twice. Find P(exactly one head). 1 There are two ways to get exactly one head: Heads-then-Tails, or Tails-then-Heads 2 P(Heads then Tails) = 0.5 \times 0.5 = 0.25 3 P(Tails then Heads) = 0.5 \times 0.5 = 0.25 4 Add the two matching paths: 0.25 + 0.25 Answer: 0.5

Example 2 — at least one red from two spins

A spinner has P(red) = 0.4 and P(not red) = 0.6 . It is spun twice, independently. Find P(at least one red). 1 It is easier to find the complement first: P(no red at all) = P(not red) \times P(not red) = 0.6 \times 0.6 = 0.36 2 P(at least one red) = 1 - P(no red at all) = 1 - 0.36 Answer: 0.64

Practice

A biased coin has P(heads) = 0.7 . It is flipped twice. Find P(exactly one head).

Your answer