Probability and conditional probability: compute simple probability as favorable over total, read probabilities from two-way frequency tables, and compute conditional probability given a restricted group.

What this skill is asking

Probability measures how likely an event is, on a scale from $0$ (impossible) to $1$ (certain). The Digital SAT (Problem-Solving and Data Analysis domain) tests simple probability (favorable over total), reading probabilities from a two-way frequency table, and conditional probability, where the question restricts attention to a particular row or column of the table before computing.

The core ideas

Probability questions hinge on choosing the correct denominator.

A worked two-way table reading

The denominator is everything.

Why the denominator changes

The difference between a simple and a conditional probability is which total sits in the denominator. A simple probability uses the whole group (the grand total). A conditional probability, signalled by "given that", "of the", or "among the", shrinks the group to a single row or column, and that subgroup's total becomes the new denominator. Reading the phrasing carefully to spot the restriction, then locating the correct subtotal in the table, prevents the most common error of dividing by the grand total when the question has restricted the population.

Reasonableness and the 0-to-1 scale

Every probability must land between $0$ and $1$, so a quick sanity check is that your fraction is in that range and matches intuition: a likely event is near $1$, a rare one near $0$. The complement rule, $P(\text{not } A) = 1 - P(A)$, is handy when "at least one" or "not" appears, since it is often easier to compute the opposite event and subtract. On two-way table questions, after computing, confirm that your numerator is a subset of your denominator (favorable outcomes are among the total you divided by); if the numerator exceeds the denominator, you have mixed up the groups.

Expected counts from a probability

The SAT also runs probability the other way: given a probability or a rate, predict a count for a larger group. If $\frac{3}{10}$ of marbles are blue and a factory makes $4000$ marbles, you expect about $\frac{3}{10} \times 4000 = 1200$ blue ones; if a survey finds that $40\%$ of a representative sample recycle, you estimate $40\%$ of the whole population recycle. This is the same proportional reasoning used for ratios: a probability is a ratio of favorable to total, so scaling it up to a population multiplies the probability by the population size. These "how many would you expect" questions reward recognising that a probability or a survey percentage can be applied to a new, larger group, provided that group is similar to the one the probability came from. As always, check the result is reasonable: an expected count cannot exceed the size of the group it is drawn from.