Probability of Exactly Two Dice Showing the Same Number

Understanding the probability of rolling exactly two of the same number on three fair dice is a classic problem in probability theory, often applicable in scenarios involving random events and outcomes. Let's explore the calculation step by step to ensure a clear understanding.

Understanding the Problem

When three fair dice are thrown, the total number of possible outcomes is derived from the combination of individual dice outcomes. Since each die has 6 faces, the total number of outcomes is 63. Let's break down the steps to find the probability:

Total Number of Outcomes

The total number of outcomes when throwing three dice is calculated as:

63 216

This means there are 216 different possible outcomes when three dice are rolled.

Favorable Outcomes

To find the number of favorable outcomes where exactly two of the three dice show the same number, the process involves several steps:

Step 1: Choosing the Number

There are 6 possible numbers on each die (1 to 6). Therefore, there are 6 ways to choose the number that appears on the two matching dice.

Step 2: Choosing Which Dices Show the Chosen Number

Next, we need to consider which two out of the three dice will show this number. There are (binom{3}{2}) 3 ways to choose 2 dice out of 3.

Step 3: Choosing the Number for the Third Die

The third die must show a different number. Since there are 5 remaining numbers (6 total - chosen number), there are 5 possibilities for the third die.

The number of favorable outcomes can be calculated as:

Number of favorable outcomes 6 (ways to choose the number) (times) 3 (ways to choose the two dice) (times) 5 (ways to choose the third die) 6 (times) 3 (times) 5 90

Calculating the Probability

The probability that exactly two of the three dice show the same number is calculated by dividing the number of favorable outcomes by the total number of outcomes:

Probability (P) (frac{text{Number of favorable outcomes}}{text{Total outcomes}}) (frac{90}{216})

Simplifying this fraction:

(frac{90}{216} frac{90 div 18}{216 div 18} frac{5}{12})

Hence, the probability that exactly two of the three dice show the same number is (boxed{frac{5}{12}}).

Alternative Formulation

For a more general case, if we consider three of the same number as including two of the same number, there are 6 ways for this to happen. Therefore, the probability would be:

(text{Probability} frac{96}{216} frac{4}{9})

Summary

In conclusion, the probability of throwing exactly two of the same number on three fair dice is ( frac{5}{12} ). This calculation demonstrates the principles of combinatorics and probability, highlighting the importance of careful enumeration and simplification in problem-solving.

Would you like to explore more such probability problems or any other related topics?