Probability (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Probability: core idea

Expected value.

**(Proof track.)** Give a complete justification. Three cards are drawn from $1$ through $8$ . Find probability their sum is even.

Hint: Specify the sample space and justify each favorable count. Answer: 1/2.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Roll two dice. Find probability their product is divisible by $6$ .
2. **(Proof track.)** Give a complete justification. A random permutation of $1,2,3,4,5$ has no fixed point. Find probability.
3. **(Proof track.)** Give a complete justification. Draw two aces-free cards from a standard deck. Find probability both are red.
4. **(Proof track.)** Give a complete justification. A fair coin is tossed until the second head. Find probability this occurs on toss four.
5. **(Proof track.)** Give a complete justification. Choose a point uniformly in a unit disk. Find probability its distance from center exceeds $1/2$ .
6. **(Proof track.)** Give a complete justification. Randomly choose a divisor of $360$ . Find probability it is a square.
7. **(Proof track.)** Give a complete justification. A box has 5 white and 5 black balls. Draw until three white. Find probability exactly four draws are needed.
8. **(Proof track.)** Give a complete justification. Two integers are independently selected from $1$ through $6$ . Find probability their gcd is $1$ .
9. **(Proof track.)** Give a complete justification. A random binary string of length six has exactly two runs. Find probability.