Probability and Statistics
What to know
Probability and statistics questions test counting, sample spaces, probability rules, and data reasoning. Compute probabilities for simple and compound events; distinguish independent from dependent events and 'and' from 'or'; use conditional probability; and apply expected value. For data, compute and interpret mean, median, mode, range, and spread; read box plots, histograms, and scatterplots; and evaluate sampling methods, bias, and the difference between association and causation.
Traps
Outliers pull the mean but barely affect the median. 'Without replacement' changes the denominator on the second draw; 'with replacement' does not. P(A and B) = P(A)P(B) only when events are independent; otherwise adjust for the changed sample space. 'Or' usually means add, subtracting overlap for non-mutually-exclusive events. A strong correlation does not prove causation, and a small or biased sample cannot support a broad claim.
Worked example
A bag holds 4 red and 6 blue marbles. Two are drawn without replacement. P(first red) = 4/10. Given a red is removed, P(second red) = 3/9. Multiply: 4/10 × 3/9 = 12/90 = 2/15. Rule of thumb: for sequential events without replacement, multiply the probabilities and decrease both the favorable count and the total after each draw.