Probability of combined events

• Sample space diagrams list every outcome of two events, such as two dice, in a grid.
• Venn diagrams show overlapping events; P(A ∩ B) is A and B, P(A ∪ B) is A or B or both.
• Tree diagrams: probabilities go beside the branches and outcomes at the ends. Multiply along a branch, add between branches.
• With replacement the probabilities stay the same on the second pick; without replacement they change.
• For “at least one”, use 1 − P(none).

A grid of every outcome

A sample space diagram is a table showing every possible outcome of two events. If each event has equally likely outcomes, so does each cell of the grid, and P = cells you want ÷ total cells.

Intersection and union

A ∩ B (the intersection) is the overlap: both A and B happen. A ∪ B (the union) is everything in A or B or both. A′ is everything outside A. A Venn diagram can hold numbers of people or probabilities; if it holds probabilities, all the regions add up to 1.

Multiply along, add between

On a tree diagram, each set of branches covers every outcome, so each set adds up to 1. Write the probability beside each branch and the outcome at the end. To find the probability of a path, multiply along it. If several paths give the event you want, add them. When the item is replaced, the second set of branches is identical to the first, because the events are independent.

The second branches change

not replaced