Probability and Statistics
Probability and Statistics tests uncertainty and data. You must use probability rules, tree diagrams and expected value, and describe data with summary statistics. Read whether events are independent or whether sampling is with or without replacement.
Probability rules
- 0 ≤ P(A) ≤ 1; P(not A) = 1 − P(A).
- P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the last term is zero.
- For independent events, P(A and B) = P(A) × P(B).
- Conditional probability: P(A | B) = P(A and B)/P(B).
- Expected value = Σ x P(x).
Statistics
- Mean = sum of values ÷ number of values; median = middle value after ordering; mode = most frequent.
- Standard deviation measures spread around the mean; a larger value means more spread.
- Correlation describes the direction and strength of a linear relationship; correlation does not prove causation.
Worked example
A bag has 5 red and 3 blue counters. Two are drawn without replacement. Find P(both red). First draw: 5/8 red. Given a red is removed, 4 red remain out of 7: P(second red | first red) = 4/7. Multiply: 5/8 × 4/7 = 20/56 = 5/14.
Traps
- Using the original total on the second draw when the item is not replaced.
- Adding probabilities for “and” events instead of multiplying.
- Finding the median without first ordering the data.
- For “at least one”, forgetting the complement method: 1 − P(none).