**Roll That Die
**A fair six-sided die is rolled 6 times. What is the probability of rolling a number greater than four at least five times?

Source: mathcontest.olemiss.edu 9/27/2010

**Solution
**Let’s start by working on a

*simpler*problem. Let’s roll the die 3 times instead of 6 times. What is the probability of rolling a number greater than four at least two times?

**STEP 1**

We first have to understand the problem. In the case of rolling a six-sided die, a number greater than 4 is either 5 or 6. At least two times means either two times or three times. So we roll the die three times and look out for either the number 5 or 6 showing either two times or three times. The following are examples of desirable outcomes:

Example 1:

Example 2:

Example 3:

Example 4:

In the above examples, . The variable occurs in the first position in **Example 1**, second position in **Example 2**, and third position in **Example 3**. And in **Example 4**, does not show up. It is much simpler to work with this variable’s position than to worry about where 5 or 6 are located.

**STEP 2
Example 1** can occur in outcomes when appears in the first position.

**Example 2**can occur in outcomes when appears in the second position.

**Example 3**can occur in outcomes when appears in the third position.

**Example 4**can occur in outcomes when there is no place for . Thus, the total number of desirable outcomes is .

**STEP 3**

The total number of possible outcomes when rolling a six-sided die three times is .

**STEP 4
**Thus, the probability of rolling a number greater than 4 at least two times is .

Now we can turn our attention to the original problem of rolling the die six times and calculate the probability of rolling a number greater than 4 at least five times.

**STEP 5
**Now the variable can occur in six different positions as follows:

**Example 1**: occurs in outcomes.

**Example 2**: occurs in outcomes.

**Example 3**: occurs in outcomes.

**Example 4**: occurs in outcomes.

**Example 5**: occurs in outcomes.

**Example 6**: occurs in outcomes.

**
**And when disappears,

**: occurs in outcomes.**

Example 7

Example 7

Thus, the total number of desirable outcomes is .

And the total number of possible outcomes is .

The probability of rolling a number greater than 4 at least five times is therefore

.