Marginal probability

A marginal probability answers a question about one variable while ignoring the other. The numerator is a row total or a column total and the denominator is always the grand total, so nobody is discarded: every individual is still counted, just sorted on one variable instead of two. The name comes from where those totals sit, out in the margins of the two-way table.

Use the same survey of 200 students that the joint probability entry tabulates: 80 are seniors and 120 are not, while 75 drive to school and 125 do not. So $P(\text{senior}) = 80/200 = 0.40$ and $P(\text{drives}) = 75/200 = 0.375$. Each of those is also the sum of the joint probabilities along its own margin, since the senior row splits into $45/200 = 0.225$ and $35/200 = 0.175$, which add to 0.40.

"The marginal probabilities are 0.40, 0.60, 0.375 and 0.625, and they add to 2, so I have made a mistake." Nothing is wrong. Marginal probabilities add to 1 within a single variable, not across the table: $0.40 + 0.60 = 1$ for class and $0.375 + 0.625 = 1$ for driving. Adding all four counts every student twice, once by year and once by transport. The set that does add to 1 across the whole table is the four joint probabilities.

A marginal probability tells you nothing about the relationship between the two variables. $P(\text{drives}) = 0.375$ is a fact about the whole school and it would read the same whether or not seniors drive more than anyone else. Seeing the relationship takes conditionals: $P(\text{drives} \mid \text{senior}) = 45/80 = 0.5625$ against $P(\text{drives} \mid \text{not senior}) = 30/120 = 0.25$.

Two-way tables and the probabilities read from them open Unit 2, at topic 2.1.