Probability and Conditional Probability

Probability measures how likely an event is, as a number from $0$ (impossible) to $1$ (certain). The SAT tests simple probability from counts, reading two-way tables, conditional probability ("given that…"), and compound events (two spins, two draws, "at least one").

• When outcomes are equally likely, probability is a fraction: favorable outcomes over total outcomes.
• "Given that" shrinks the group you choose from. The denominator becomes the size of the given group, not the grand total.
• In a two-way table, identify the denominator first: the whole table, one row, or one column.
• For independent events (spins, coin flips, draws with replacement), multiply the probabilities.
• Without replacement, the second probability changes because one item is gone.
• "At least one" is usually easiest as $1 - P(\text{none})$.
• When the data are given as percents, imagine a convenient total (such as $1{,}000$ people) and convert to counts.

$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \qquad P(\text{not } A) = 1 - P(A)$

$P(A \mid B) = \frac{\text{number in both } A \text{ and } B}{\text{number in } B}$

Example: a box has $5$ red and $7$ white tiles. Drawing two red tiles without replacement has probability $\frac{5}{12} \cdot \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$.

A jar has $6$ lemon, $9$ cherry, and $5$ grape candies. One is chosen at random. What is the probability it is not cherry?

1. Total: $6 + 9 + 5 = 20$.
2. Not cherry: $6 + 5 = 11$.
3. Probability: $\frac{11}{20}$.

Check: $P(\text{cherry}) = \frac{9}{20}$, and $1 - \frac{9}{20} = \frac{11}{20}$. Correct.

A store surveyed customers about where they shop.

If a customer who shops online is chosen at random, what is the probability the customer is under 30?

1. "Who shops online" is the given group, so the denominator is the Online column total, $75$.
2. Under 30 and online: $48$.
3. Probability: $\frac{48}{75} = \frac{16}{25} = 0.64$.

Compare: the probability that a customer under 30 shops online is $\frac{48}{80} = 0.6$. Same cell, different denominator. That is a favorite SAT trap.

• Using the grand total for a conditional probability. $\frac{48}{170}$ answers "online and under 30," not "under 30, given online."
• Swapping the condition. $P(A \mid B)$ and $P(B \mid A)$ share a numerator but have different denominators.
• Forgetting to reduce the total without replacement. After one item is drawn, there are $n - 1$ left.
• Adding instead of multiplying for "this and then that."
• Counting "at least one" case by case and missing a case. The complement is faster and safer.
• Taking a percent of the wrong group. "25% of the coffee drinkers" is 25% of that group, not of everyone.