SAT® · Math: Problem-Solving and Data Analysis
Area Models and Other Probability Representations
An area model represents a two-stage probability as a rectangle cut from a unit square: one cut for the first event, a second cut (inside the first) for the second, and the shaded area is their product.
Picture a square with side 1, standing for the whole shipment or population. Slice its width into two pieces at the percent for event A: say 40 percent red, 60 percent not red. Now take just the red strip and slice its height at the percent for event B given that you are already in that strip: 30 percent of the red items have stripes. The little rectangle sitting inside the red strip, 30 percent of its height, has area 0.40 x 0.30 = 0.12. That rectangle is both red and striped, so 12 percent of the whole shipment is both red and striped. Area makes this obvious: you are not adding two numbers, you are finding the overlap, and overlap on a picture is literally a smaller region sitting inside a bigger one.
The same picture extends naturally. If a factory's output splits between two machines, each machine's strip has its own width (share of output) and its own defect rate carves out a smaller rectangle inside that strip. Add the two rectangles' areas and you get the total defective percent for the whole factory, because the two strips cover everything and don't overlap. This is the same logic as a two-way table: rows and columns of a table are just a grid version of the same cuts, and a tree diagram is the same idea written as branches instead of area, with probabilities multiplying along each path.
The model also flexes when the second event's chance depends on the outcome of the first, which is exactly conditional probability: the height of the second cut is allowed to change from one vertical strip to the next, because 'percent striped among red items' need not match 'percent striped among non-red items'. That flexibility is what separates this from simple independent multiplication and links the topic to conditional probability problems elsewhere in data analysis.
Key terms
- area model
- A unit square whose cuts represent probabilities, where a region's area equals the chance of the events it represents.
- joint probability
- The chance that two events both happen, shown as the overlapping rectangle in an area model.
- conditional probability
- The chance of one event given that another has already occurred, shown as a cut whose size can change depending on which strip it's inside.
- two-way table
- A grid of counts or percents sorted by two categories at once, the tabular version of an area model.
Watch it explained
- How it worksHow does slicing a square twice turn two percentages into one combined probability?
- Worked exampleFind the percent of items that are red and striped when 40 percent are red and 30 percent of those have stripes.
- Going furtherWhat changes in an area model when the second event depends on how the first one turned out?
Check yourself
End of topic.