2021 AMC 10B Problems/Problem 18
A fair -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
There is a chance that the first number we choose is even.
There is a chance that the next number that is distinct from the first is even.
There is a chance that the next number distinct from the first two is even.
, so the answer is
Every set of three numbers chosen from has an equal chance of being the first 3 distinct numbers rolled.
Therefore, the probability that the first 3 distinct numbers are is
Note that the problem is basically asking us to find the probability in some permutation of that we get the three even numbers in the first three spots.
There are ways to order the numbers and ways to order the evens in the first three spots and the odds in the next three spots.
Therefore the probability is .