The probability that the game ends after all three players have had their turns is 1/8.

Here's the reasoning:

1. For the game to end after all three players have had their turns, it means that none of them won during their respective turns.
2. The probability of Alice not winning in her turn is 1 - 1/2 = 1/2.
3. The probability of Bob not winning in his turn is 1 - 1/3 = 2/3.
4. The probability of Carol not winning in her turn is 1 - 1/4 = 3/4.
5. Since these events are independent, the probability of all three events happening simultaneously is the product of their individual probabilities: (1/2) * (2/3) * (3/4) = 1/8.

Thus, the probability that the game ends after all three players have had their turns is 1/8.
In a game, each player gets one turn. There are three players: Alice, Bob, and Carol. The probability of Alice winning in her turn is 1/2, the probability of Bob winning in his turn is 1/3, and the probability of Carol winning in her turn is 1/4. What is the probability that the game ends after all three players have had their turns?