Probability Trees
Year 10 (IGCSE) 📈 Statistics & Probability Draw and use probability tree diagrams for combined events.
🌳 Tree Diagrams
A probability tree diagram shows all possible outcomes of combined events, with probabilities on each branch.
⚡ Key Rules
Along branches (AND): multiply probabilities.Between branches (OR): add probabilities from different routes to the same outcome.
All probabilities leaving any single node must sum to 1.
🔗 Independent vs Dependent Events
Events are independent if one does not change the probability of the other. They are dependent if the first outcome affects subsequent probabilities.
Independent (with replacement): Bag: 3 red, 2 blue. Draw, replace, draw again. $P(\text{red, red}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}$
Dependent (without replacement): Same bag, no replacement. $P(\text{red, red}) = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$
📊 Conditional Probability
Conditional probability is the probability of an event given that another event has already occurred.
⚡ Conditional Probability Formula
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$Example: In a group of 20: 12 study French, 8 Spanish, 5 study both. Given a student studies French, the probability they also study Spanish $= \frac{5}{12}$.
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Interactive Demonstration — Probability Trees
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Interactive demonstration available in the Calculator tab!
🌳 Probability Tree Builder
Two-stage probability tree — events A and B.