Probability

The basic formula:

$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{number of total outcomes}}$

Probabilities are always between 0 and 1. $P = 0$ means impossible; $P = 1$ means certain.

Reading two-way tables. Most SAT probability problems give you a table like:

The key skill: identify what the denominator is. Different questions use different denominators:

• "What is the probability that a randomly selected student passes?" → $\frac{65}{80}$ (whole table is the population).
• "Given that a student is a girl, what's the probability she passes?" → $\frac{35}{40}$ (only girls is the population).
• "Given that a student passed, what's the probability she's a girl?" → $\frac{35}{65}$ (only passers is the population).

This is conditional probability — given X, what's the probability of Y — and the SAT loves it.

The 'and' / 'or' rules. Less commonly tested but worth knowing:

• $P(A \text{ and } B) = P(A) \cdot P(B)$ if events are independent.
• $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

Complement rule. $P(\text{not } A) = 1 - P(A)$. Useful when computing not at all probabilities.

"Probability that at least one of three students passes." → easier as $1 - P(\text{all three fail})$.

Common SAT trick: Q phrased as "of those who passed, what fraction were girls?" — this isn't really probability language but the calculation is the same: count the girls who passed, divide by all who passed.

Tip: circle the given condition in the question. That tells you the denominator. Underline what's being asked. That tells you the numerator.