Statistics and Probability: Tables, Conditional Probability, and Data Reasoning
What this domain tests
Statistics and Probability asks you to draw conclusions from data rather than to compute in the abstract. Expect two-way frequency tables, conditional probability and independence, sample spaces and simulation, the design of studies (census, sample survey, experiment, observational study), the normal distribution and standard deviation, lines of best fit, correlation, residuals, and the difference between association and causation.
Core rules
- Two-way tables separate joint frequencies (a single cell) from marginal frequencies (row and column totals). The whole table sums to the grand total.
- Conditional probability restricts the sample space: P(A | B) = P(A and B)/P(B). The denominator is always the total for the given condition, not the grand total.
- Independence holds when P(A | B) = P(A), equivalently P(A and B) = P(A)·P(B). With counts, compare a cell's proportion to the product of its marginal proportions.
- Study design: random sampling supports generalization to a population; random assignment to treatment groups supports causal claims. An observational study can reveal association but not causation.
- Standard deviation measures typical distance from the mean. It is never negative and is strongly inflated by outliers, which barely move the median.
- Regression: the line of best fit minimizes squared residuals. The correlation coefficient r lies between −1 and 1, and r² is the fraction of variation in the response explained by the model. A curved residual plot signals that a line is the wrong model.
Worked example
Of 120 students, 50 play an instrument; of those, 30 also play a sport. Of the 70 who do not play an instrument, 40 play a sport.
Find the probability a student plays a sport, given that the student plays an instrument. The condition limits us to the 50 instrument players, and 30 of them play a sport, so P(sport | instrument) = 30/50 = 0.60. For contrast, P(sport) = 70/120 ≈ 0.583. Dividing by 120 instead of 50 would produce 0.25, the joint probability P(instrument and sport), not the conditional probability requested.
Traps that cost marks
- Using the grand total as the denominator of a conditional probability.
- Reversing P(A | B) and P(B | A), which are generally unequal.
- Claiming causation from an observational study or from a strong correlation.
- Predicting far outside the observed data range with a regression line.
- Assuming a larger mean implies a larger standard deviation.