Probability of Rolling at Least Two Matching Dice

When rolling three dice, a common question is whether at least two of the dice will show the same number. This article explores the probability of this event occuring, using both direct and complementary probability approaches.

Total Number of Outcomes

When three standard six-faced dice are rolled, the total number of possible outcomes is:

Total Outcomes 63 216

Favorable Outcomes - Direct Approach

To find the probability of at least two dice showing the same number, we first calculate the number of favorable outcomes.

Choosing the Matching Pair

Select the number that appears on the two matching dice. There are 6 possible choices (1 through 6).

Choosing the Non-Matching Die

Choose the face of the third die, ensuring it's different from the first. There are 5 such choices.

Choose which two of the three dice show the matching number. There are:

32 3 ways to do this.

Total Favorable Outcomes

The total number of favorable outcomes is:

6 × 5 × 3 90

Calculating Probability

The probability of rolling at least two matching dice is:

Probability 90 / 216 5 / 12 ≈ 41.67%

Complementary Probability Approach

Alternatively, we can use the complementary probability approach to find the probability that at least two of the three dice show the same number. This involves calculating the probability that all three dice show different numbers and then subtracting that from 1.

Total Outcomes

As before, the total number of possible outcomes is:

6 × 6 × 6 216

Favorable Outcomes for All Different Numbers

To find the number of outcomes where all three dice show different numbers:

The first die can show any of the 6 numbers. The second die can show any of the remaining 5 numbers to be different from the first. The third die can show any of the remaining 4 numbers to be different from the first two. Thus, the number of ways to roll three dice with all different numbers is:

6 × 5 × 4 120

Probability of All Different Numbers

The probability that all three dice show different numbers is:

P(all different) 120 / 216 5 / 9

Probability of At Least Two Dice Being the Same

The probability of at least two dice showing the same number is:

P(at least two the same) 1 - P(all different) 1 - 5 / 9 4 / 9 ≈ 44.44%

Conclusion

In conclusion, the probability that at least two of the three dice rolled are the same number is:

4 / 9 or approximately 44.44%