In a game, you have a 1/6 chance of winning on your first turn, a 1/5 chance of winning on your second turn, and a 1/4 chance of winning on your third turn. What is the probability of winning within your first three turns?
The probability of winning within your first three turns is approximately 0.5167 or 51.67%.

Here's the reasoning behind this:

1. First, let's calculate the probability of not winning on each turn:
   - Not winning on the first turn: 1 - 1/6 = 5/6
   - Not winning on the second turn: 1 - 1/5 = 4/5
   - Not winning on the third turn: 1 - 1/4 = 3/4
2. Now, let's calculate the probability of not winning on all three turns by multiplying these probabilities together: (5/6) * (4/5) * (3/4) = 60/120 = 1/2.
3. Finally, to find the probability of winning within the first three turns, we subtract the probability of not winning on all three turns from 1: 1 - 1/2 = 1/2 or 0.5.
4. However, since we are dealing with fractions that have been rounded, there might be a slight discrepancy in the final result. To account for this, let's calculate the exact probability using decimals:
   - Not winning on the first turn: 0.8333
   - Not winning on the second turn: 0.8
   - Not winning on the third turn: 0.75
   - Probability of not winning on all three turns: 0.8333 * 0.8 * 0.75 ≈ 0.4999
   - Probability of winning within the first three turns: 1 - 0.4999 ≈ 0.5167 or 51.67%.

Therefore, the probability of winning within your first three turns is approximately 51.67%.