Probability
Probability tests the language of chance, sample-space diagrams, tree diagrams and combined events. Probabilities lie from 0 to 1, and the probabilities of all mutually exclusive outcomes sum to 1.
Rules
- P(not A) = 1 − P(A).
- For mutually exclusive events, add: P(A or B) = P(A) + P(B).
- For independent events, multiply: P(A and B) = P(A) × P(B).
- With tree diagrams, multiply along a branch and add between branches.
- Without replacement, the denominator and numerator change for the second event.
Worked example
A bag has 5 red and 3 blue counters. Two counters are taken without replacement. Find P(one red and one blue in either order). First pick red, second blue: 5/8 × 3/7 = 15/56. First blue, second red: 3/8 × 5/7 = 15/56. These are mutually exclusive, so add: 30/56 = 15/28. The second denominator is 7 because one counter has gone.
Traps
Using 8 again for the second draw without replacement; adding the probabilities of independent events instead of multiplying; forgetting the two orders in “one of each”; assuming outcomes are equally likely when they are not; giving a probability greater than 1.