Three Dice Are Tossed: What is the Probability That the Numbers Shown Will All Be Different?

To find the probability that all numbers shown on three tossed dice will be different, we can approach it step by step.

First, we need to know the total number of possible outcomes when rolling three dice. Since each die has 6 faces, the total number of possible outcomes is:

Total Outcomes = 6 × 6 × 6 = 216

Next, we need to determine the number of favorable outcomes where all three dice show different numbers. To ensure all numbers are different:

1. For the first die, you can land on any of the 6 faces.
2. For the second die, to ensure it is different from the first, you have 5 remaining choices.
3. For the third die, to ensure it is different from both the first and second, you have 4 remaining choices.

Thus, the number of favorable outcomes where all three dice show different numbers is:

Favorable Outcomes = 6 × 5 × 4 = 120

Now, we can find the probability by dividing the number of favorable outcomes by the total number of outcomes:

Probability = Favorable Outcomes / Total Outcomes = 120 / 216

This fraction can be simplified:

Probability = 120 / 216 = 5 / 9

Therefore, the probability that all numbers shown on the three tossed dice will be different is:

5/9