Joint Probability vs Marginal Probability
Both terms below come up in the same part of the course, and students mix them up. Here is each one defined on its own, side by side, so you can see where they part company.
Joint probability
Probability
A joint probability is the chance that two events both happen, written P(A and B), so the same individual or trial has to meet both conditions.
A joint probability describes the intersection , so both conditions have to be met by the same individual or trial. In a two-way table you find it by dividing an interior cell count by the grand total, never by a row or column total. Among 200 surveyed students, 45 are seniors who drive to school, so . Adding all the joint probabilities across a row gives the marginal probability for that row.
Marginal probability
Probability
A marginal probability is the probability of one event on its own, read from a row or column total in the margins of a two-way table.
It answers a question about a single variable while ignoring the other one, which is why the totals live in the margins of the table. You divide a row or column total by the grand total, so the denominator is always the full sample size. Among 200 surveyed students, 80 are seniors, so . That value also equals the sum of the joint probabilities in the senior row, since those cells split the row total into pieces.