Question

A standard pair of six-sided dice are rolled. What is the probability of rolling a sum less than or equal to $10$ ?

Answer 1

Find the probability of rolling a sum less than or equal to $10$.

Step 1: Find the number of possible outcomes of six-sided dice.

Given that, the probability is $10$.

A standard pair of $6$ sided dice are rolled. the possible outcomes are:

$$\begin{gathered} (1,1),(1,2),(1,3),(1,4),(1,5),(1,6) \\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6) \\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6) \\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6) \\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6) \\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6). \end{gathered}$$

So, there are $36$ total outcomes.

Step 2: Find the probability of rolling a sum less than or equal to $\mathbf{10}.$

A number of outcomes which has a sum less than or equal to $10$ are:

$$\begin{gathered} (1,1),(1,2),(1,3),(1,4),(1,5),(1,6) \\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6) \\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6) \\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6) \\ (5,1),(5,2),(5,3),(5,4),(5,5) \\ (6,1),(6,2),(6,3),(6,4). \end{gathered}$$

So, there are $33$ outcomes that have the sum of $10$ or less than $10$.

Use the probability formula: $\mathrm{P}\left(\mathrm{A}\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{successful}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{possible}\mathrm{outcomes}}$

Here, the number of successful outcomes is $33$.

The total number of possible outcomes $36$.

Substitute the values in the formula:

$\begin{array}{rcl}\mathrm{P}\left(\mathrm{A}\right)& =& \frac{33}{36}\\ & =& \frac{11}{12}\end{array}$

Hence, the probability of rolling a sum less than or equal to $10$ in a standard pair of six-sided dice is $\frac{\mathbf{11}}{\mathbf{12}}.$