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Exactly vs at least vs at most

Many probability mistakes come from reading the event wording too loosely. The phrase tells you which outcomes should be included.

Exactly

One target count only

Exactly 3 heads means only the outcomes with 3 heads count. Nothing with 2 or 4 heads belongs in the event.

At least

The target count and everything above it

At least 1 head means 1, 2, 3, and so on up to the maximum possible count. It is often fastest to use the complement, such as 1 minus the probability of 0 heads.

At most

The target count and everything below it

At most 2 aces means 0, 1, or 2 aces. This wording creates a range, not a single exact count.

Decision rule

Write out the allowed counts before you calculate

Before touching a formula, list the success counts that satisfy the event. If the list has one number, it is an exact event. If it has several numbers, you are summing a range or using a complement.