Suppose a six-sided die has sides numbered through . If a person throws the die two times, what is the probability that the second number will be larger than the first?

Source: NCTM Mathematics Teacher, September 2006

**Solution
**The desirable outcomes are

where is the first number and are the second number

Total number of desirable outcomes

Number of possible outcomes

Probability that the second number is larger than the first =

**Answer**:

**Alternative solution**

First, the die is thrown and the first number comes up each with a probability equal . Then, the die is thrown again and we want the second number larger than the first.

probability =

probability =

probability =

probability =

probability =

probability =

Probability of rolling the die two times and the second number is larger than the first