Statistics, Probability, and Discrete Mathematics
What to know
Statistics: measures of center (mean, median, mode), spread (range, interquartile range, standard deviation conceptually), reading and interpreting data displays (histograms, box plots, scatterplots), and the effect of outliers on the mean versus the median. Probability: sample spaces, counting principles, independent vs. dependent events, conditional probability, and expected value. Discrete topics include combinations/permutations, basic graph ideas, and logical reasoning.
Traps to watch
- Mean is pulled by outliers; median is resistant. If a question mentions skew or extreme values, lean on the median.
- "And" vs. "or": independent events multiply ($P(A \text{ and } B) = P(A)P(B)$); mutually exclusive events add.
- Without replacement changes the denominator and often the numerator on the second draw.
- Permutations vs. combinations: order matters → permutations; order doesn't → combinations.
- Expected value is a weighted average: multiply each outcome by its probability and sum.
- Median with an even count is the average of the two middle values.
Worked example
A bag holds 4 red and 6 blue marbles. Draw two without replacement. Probability both are red: $P = \frac{4}{10} \cdot \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$. Note the second fraction uses 3 red out of 9 remaining - the classic trap is keeping the denominator at 10. Rule of thumb: after each draw, update both the favorable count and the total.