Branching Out
Two events, one diagram: use probability trees to combine events the right way — multiply along the branches, add between them, and watch what changes when you do not replace.
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Branching Out 🌳
When two things happen — two coin flips, two counters drawn — a **probability tree** lays out every combined outcome so nothing is missed. Two rules run the whole topic: **multiply** probabilities **along** a path, and **add** the paths that satisfy your event. Master those and combined probability is yours.
How a tree works 🔀
Each **branch** shows one outcome and its probability. To find the chance of a full result: • **Multiply along** the branches of a path (first event, then second). • **Add** the probabilities of the different paths that count as your event. A useful check: the branches coming from any single point always **add up to 1**.
Read the tree
An interactive activity.
Independent events ✖️
Two events are **independent** when one does not affect the other (a coin has no memory). Then: **P(A and B) = P(A) × P(B).** For a fair coin, P(Heads) = 0.5, so P(Heads and Heads) = 0.5 × 0.5 = **0.25** — exactly the "multiply along the path" rule.
Heads then heads
An interactive activity.
Order the method
An interactive activity.
Win both spins
An interactive activity.
Multiply or add?
To find the probability of one specific outcome, such as "Heads then Heads", what do you do with the branch probabilities along that path?
- Multiply them together
- Add them together
- Subtract one from the other
- Take the larger of the two
True about trees
Pick the TWO statements that are correct about probability tree diagrams.
- You multiply the probabilities along a path
- The branches from any single point add up to 1
- You add the probabilities along a single path
- A branch can have a probability greater than 1
At least one ➕
"At least one" questions are quickest through the **opposite**: **P(at least one) = 1 − P(none).** For two coin flips, P(at least one Head) = 1 − P(no Heads) = 1 − P(Tails, Tails) = 1 − 0.25 = **0.75**. Finding "none" first is far less work than adding every other case.
At least one head
An interactive activity.
Without replacement 🚫
If an item is **not replaced**, the second probability **changes** — there are fewer items left. A bag has 3 red and 2 blue counters (5 total). P(first red) = 3/5. If you keep it, only 4 counters remain with 2 red, so P(second red) = 2/4. This shift is the classic grade-9 trap — never reuse the first probability.
Two reds
An interactive activity.
The tree rules
On a probability tree, _____ the probabilities along a path to find one combined outcome, and _____ the results of the different paths that satisfy the event. The branches from any single point always add up to _____.