When three dice are tossed, what is the probability that all the numbers shown are different?

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Understanding the Probability of Rolling Three Different Numbers on Dice

When you toss three dice, there are a total of 6 faces on each die, resulting in 6 imes 6 imes 6 = 216 possible outcomes.

Calculating Favorable Outcomes

To find the probability that all three numbers show different faces, we first calculate the number of favorable outcomes:

  • For the first die, you can roll any of the 6 faces.
  • For the second die, you can only roll 5 different faces (since it must be different from the first die).
  • For the third die, you can roll 4 different faces (since it must be different from both the first and second dice).

This gives us a total number of favorable outcomes calculated as follows:

Favorable Outcomes = 6 imes 5 imes 4 = 120

Calculating the Probability

The probability (P) that all numbers rolled are different can be calculated using the formula:

P(All Different) = rac{Favorable ext{ }Outcomes}{Total ext{ }Outcomes}

Using our numbers:

P(All Different) = rac{120}{216}

This fraction simplifies to:

P(All Different) = rac{5}{9}

Conclusion

Thus, the probability that when tossing three dice, all the numbers shown will be different is rac{5}{9}. This means there is a good chance that each die will show a unique number!

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