Statistics and Probability
The Statistics and Probability domain tests how you summarize data, read two-way tables, interpret scatterplots and lines of fit, and reason about probability. MCAS Grade 10 emphasizes conditional probability, independence, sampling, and margin of error.
Core rules
- Two-way tables: joint frequencies are in the body, marginal frequencies are totals, and conditional frequencies use a row or column total as the denominator.
- Conditional probability: P(A | B) = P(A and B)/P(B). The vertical bar means given that B occurred.
- Independence: events A and B are independent if P(A and B) = P(A)·P(B), equivalently P(A | B) = P(A).
- Line of best fit: slope is the predicted change in y per one-unit increase in x; the y-intercept is the predicted y when x = 0. Correlation does not prove causation.
- Sampling: random sampling reduces bias; larger samples generally produce a smaller margin of error. Margin of error describes uncertainty, not a guarantee.
- Normal distributions: roughly 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
Worked example
In a group of 100 students, 40 take art, 30 take music, and 12 take both. Find P(art | music) and decide whether the events are independent. Given music, the denominator is the 30 music students: P(art | music) = 12/30 = 0.4. Check independence: P(art)·P(music) = 0.4·0.3 = 0.12, which equals P(both) = 12/100. The events are independent.
Traps
- Reversing a conditional probability: P(A | B) is not P(B | A).
- Using the grand total instead of the given row or column total.
- Treating a strong correlation as proof of cause and effect.
- Extrapolating a line of best fit far beyond the data.