Understanding the Probability of Rolling a 5 or 6 on a Pair of Fair Dice

When it comes to playing games or simply performing mathematical exercises, understanding the probability of various outcomes, such as rolling a 5 or 6 on a pair of fair dice, is crucial. This article aims to provide a comprehensive explanation on how to calculate this probability.

The Basics of Fair Dice

A pair of fair six-sided dice means each die has six faces, each with a number between 1 and 6. They are equally likely to land on any one of these faces, ensuring the dice are fair and unbiased.

Calculating the Total Number of Possible Outcomes

When rolling a pair of dice, the total number of possible outcomes can be calculated by multiplying the number of outcomes on one die by the number of outcomes on the other die.

Total Outcomes 6 (faces on die 1) × 6 (faces on die 2) 36 possible outcomes

Favorable Outcomes for Rolling a 5 or 6

To determine the probability of rolling a 5 or 6, we need to count the number of favorable outcomes for each of these numbers.

Rolling a 5

The combinations that yield a total of 5 are:

1 4 2 3 3 2 4 1

So, there are 4 combinations that result in a total of 5.

Rolling a 6

The combinations that yield a total of 6 are:

1 5 2 4 3 3 4 2 5 1

Therefore, there are 5 combinations that result in a total of 6.

Total Favorable Outcomes

By adding the number of favorable outcomes for a total of 5 and a total of 6, we get the total number of favorable outcomes:

Total Favorable Outcomes 4 (for total of 5) 5 (for total of 6) 9 favorable outcomes

Calculating the Probability

The probability of rolling either a 5 or a 6 is given by the ratio of favorable outcomes to the total outcomes. This can be calculated as:

P(5 or 6) Favorable Outcomes ÷ Total Outcomes

P(5 or 6) 9 ÷ 36 1 ÷ 4 0.25

Conclusion

The probability of rolling either a 5 or a 6 on a pair of fair dice is 1/4 or 0.25. Understanding this basic principle can help in various scenarios, from casual games to more complex statistical analyses.

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