Probability Theory and Statistics
This domain tests basic probability rules for single and combined events, plus the interpretation of mean, median, mode, spread, and the normal distribution.
Probability rules
- P(E) = favorable outcomes ÷ total outcomes, and 0 ≤ P(E) ≤ 1.
- Complement: P(not E) = 1 − P(E).
- Addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events the overlap is zero.
- Independent events: P(A and B) = P(A) · P(B).
- Sampling without replacement changes the total (and sometimes the favorable count) after each draw, so multiply conditional fractions: P(first) × P(second given first).
- Conditional probability: P(B | A) = P(A ∩ B)/P(A).
- Expected value = Σ(value × probability).
Statistics
- Mean = sum ÷ count; it responds strongly to outliers. Median = middle value of an ordered list (average the two middle values when the count is even); it resists outliers. Mode = most frequent value.
- Range = maximum − minimum. Standard deviation measures typical distance from the mean and carries the same units as the data.
- Normal distribution: roughly 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. A z-score z = (x − μ)/σ says how many standard deviations a value sits from the mean.
Worked example
A bag holds 5 red and 3 blue marbles (8 total). Two are drawn without replacement.
- P(both red) = (5/8)(4/7) = 20/56 = 5/14 ≈ 0.357. The second denominator is 7 because one marble is gone, and the numerator drops to 4 because a red one was removed.
- P(at least one blue) = 1 − 5/14 = 9/14 ≈ 0.643.
- Test scores 72, 85, 85, 90, 98: mean = 430/5 = 86; median = 85; mode = 85. The high score of 98 pulls the mean above the median.
Traps that cost marks
- Leaving the denominator unchanged on a without-replacement second draw.
- Adding probabilities for events that can occur together without subtracting the overlap.
- Treating independent events as mutually exclusive, or the reverse.
- Forgetting to average the two middle values for an even number of data points.
- Reporting a median that was never computed from sorted data.