When two or more events happen together, we combine probabilities using the addition and multiplication rules, often with the help of a tree diagram.
The two rules
- Multiplication ('AND'): for independent events, P(A and B) = P(A) × P(B).
- Addition ('OR'): for mutually exclusive events, P(A or B) = P(A) + P(B).
Key idea
On a tree diagram, multiply along the branches (AND) and add between different paths (OR). The probabilities on branches from one point add up to 1.
With and without replacement
If an item is replaced, the totals stay the same each time, so the two picks are independent. If it is not replaced, both the numerator and denominator decrease for the second pick, and the second probability then depends on what happened first — this is a conditional probability. Reading along the second set of branches of a tree diagram gives P(second event given the first).
Worked example
A bag has 3 red and 2 green balls. Two are drawn without replacement. Find P(both red).
P(1st red) = 3/5. After removing one red, 2 red remain out of 4: P(2nd red) = 2/4.
P(both red) = 3/5 × 2/4 = 6/20 = 3/10.
A tree diagram is the safest method for two-stage problems. Write the probability on every branch, multiply along each complete path to get the probability of that combination, and then add the probabilities of all the paths that satisfy what the question asks. As a check, the probabilities of all the final paths should add up to 1. For an 'at least one' question it is usually quickest to work out P(none) first and subtract from 1.
Remember
- Independent 'AND' → multiply.
- Mutually exclusive 'OR' → add.
- Without replacement changes the second denominator.