**Middle School Lesson 73: Probability with Dice**

We have flipped coins (

**ProbCoin**), and now we will roll the dice. This concept appeared

**10 times**last year, with a median placement at

**question # 73**.

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With dice problems on number sense tests, you can always assume that you are dealing with standard 6-sided dice (I've never seen anything else). The probability of rolling a certain number (between 1 and 6, inclusive) when rolling one die is

**1/6**. When rolling two dice, there are 36 possible outcomes:

**1 & 1 1 & 2 1 & 3 1 & 4 1 & 5 1 & 6**

2 & 1 2 & 2 2 & 3 2 & 4 2 & 5 2 & 6

3 & 1 3 & 2 3 & 3 3 & 4 3 & 5 3 & 6

4 & 1 4 & 2 4 & 3 4 & 4 4 & 5 4 & 6

5 & 1 5 & 2 5 & 3 5 & 4 5 & 5 5 & 6

6 & 1 6 & 2 6 & 3 6 & 4 6 & 5 6 & 6

2 & 1 2 & 2 2 & 3 2 & 4 2 & 5 2 & 6

3 & 1 3 & 2 3 & 3 3 & 4 3 & 5 3 & 6

4 & 1 4 & 2 4 & 3 4 & 4 4 & 5 4 & 6

5 & 1 5 & 2 5 & 3 5 & 4 5 & 5 5 & 6

6 & 1 6 & 2 6 & 3 6 & 4 6 & 5 6 & 6

**1**result out of 36 =

**1/36**

Rolling a 3:

**2**results out of 36 = 2/36 =

**1/18**

Rolling a 4:

**3**results out of 36 = 3/36 =

**1/12**

Rolling a 5:

**4**results out of 36 = 4/36 =

**1/9**

Rolling a 6:

**5**results out of 36 =

**5/36**

Rolling a 7:

**6**results out of 36 = 6/36 =

**1/6**

Rolling an 8:

**5**results out of 36 =

**5/36**

Rolling a 9:

**4**results out of 36 = 4/36 =

**1/9**

Rolling a 10:

**3**results out of 36 = 3/36 =

**1/12**

Rolling an 11:

**2**results out of 36 = 2/36 =

**1/18**

Rolling a 12:

**1**result out of 36 =

**1/36**

**How to Solve:**

For number sense questions asking the probability of rolling a certain sum with 2 dice:

- If the sum is 7 or less:
**(sum - 1)/36** - If the sum is 7 or greater:
**(13 - sum)/36**

Example 1: The probability of getting a 3 when rolling a single die is ___

Example 1: The probability of getting a 3 when rolling a single die is ___

- The answer is
**1/6**.

**Example 2: The probability of rolling at least a 5 when rolling a single die is ___**

- There are two ways to roll at least a 5: 5 or 6.
- The answer is 2/6, which is
**1/3**.

**Example 3: The probability of getting a sum of 9 when rolling a pair of dice is ___**

- The answer is (13 - 9)/36 = 4/36 =
**1/9**.

**Example 4: The probability of getting at least a 9 when rolling a pair of dice is ___**

- For the numerator, think of the ways you can get a 9, 10, 11, or 12.
- The answer is (4 + 3 + 2 + 1)/36 = 10/36 =
**5/18**.

**Example 5: The probability of rolling a different number on each die when rolling two 6-sided dice is ___**

- There are 6 ways you can roll the
**same**number on two dice: 1 & 1, 2 & 2, 3 & 3, 4 & 4, 5 & 5, and 6 & 6. - That leaves
**30 ways**you can roll a different number. The answer is 30/36 =**5/6**.

**Check back soon for a free worksheet to help you practice ProbDice.**

**Up Next for Middle School: SetsIntersect**