Probability and Statistics
Probability and Statistics tests counting, basic and conditional probability, and the summary and interpretation of data sets.
Core rules
- Probability P(A)=favourable outcomes/total equally likely outcomes; 0≤P(A)≤1. P(A′)=1−P(A).
- Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B). For mutually exclusive events the intersection term is zero.
- Multiplication rule: P(A∩B)=P(A)P(B|A). Events are independent when P(A∩B)=P(A)P(B).
- Permutations count ordered arrangements: nPr=n!/(n−r)!. Combinations count selections: nCr=n!/(r!(n−r)!).
- Mean = sum/n. Median is the middle value after ordering. Mode is the most frequent value. Range = max−min. For population variance, divide squared deviations by n; for sample variance, divide by n−1.
Worked example
Two fair dice are rolled. Find P(sum>9). The sample space has 36 equally likely pairs. Sums greater than 9 are 10, 11, 12. Count: sum 10 has 3 pairs, sum 11 has 2, sum 12 has 1, total 6. So P=6/36=1/6.
Traps
“At least one” is usually easier as 1−P(none). With draws without replacement, denominators decrease and outcomes are not independent. Do not use the mean to describe skewed data without noting the median. In grouped data, use class midpoints, and remember that the median class is found from cumulative frequency, not from the highest frequency. Distinguish population and sample standard deviation.