Probability (Hard)

Late AMC 10/12 through early AIME — dense computation.

Probability: core idea

Conditional.

Two fair dice are rolled. Find $P(\text{prime sum})$ .

Hint: Choose an equally likely sample space and count favorable outcomes. Answer: 5/12.

1. From $4R,3B,2G$ marbles draw two without replacement. Find $P(\text{same color})$ .
2. A fair coin is tossed five times. Find $P(\text{exactly 3 heads})$ .
3. Select an integer from $1$ to $100$ . Find $P(3\mid n\text{ or }5\mid n)$ .
4. Draw two cards from $1$ through $10$ . Find $P(\text{even product})$ .
5. A random point is in a $4\times4$ square. Find $P(\text{within 2 of center})$ .
6. Spinner sectors are $1,2,2,3,3,3$ . Spin twice: find $P(\text{sum }5)$ .
7. Three fair dice are rolled. Find $P(\text{all distinct})$ .
8. Randomly arrange $A,B,C,D$ . Find $P(A,B\text{ have one letter between})$ .
9. From $6$ good and $2$ defective bulbs, find $P(\text{first defective is third})$ .