Finding an Unknown Frequency from a Probability

Probability is just a frequency divided by a total: P = count / total. Every problem on this topic hands you two of those three pieces and asks for the third. Usually you get a probability and a total, so you multiply them to get the missing count. The grand total is often stated directly, but sometimes it's a row or column total in a two-way table, so you have to read the display carefully to know which total the probability is actually built on.

The mechanism is always the same multiplication, but the setup shifts. In a two-way table, a probability might describe one specific cell, like 'row 1 and column 1.' If you also know the row total, you find that one cell by multiplying, then subtract from the row total to get the other cell in that row. Sometimes two cells are unknown and linked by a single total, so finding one unlocks the other through subtraction rather than a second multiplication. When the red count is given and the probability of red is a variable x, flip the formula: total = red count / x, then blue = total minus red count.

This sits next to percent change and basic ratios in the same unit, because all three ask you to move fluently between fractions, percents, and raw counts. It also feeds directly into conditional probability, where you restrict attention to one row or column of a table before computing a new probability. Getting comfortable finding a missing frequency here makes those conditional setups much faster, since you already know how to read totals correctly off a table.