Question

Two dice having six sides with numbers $1,2,3,4,5$ and $6$ written on them or rolled. answer the questions that follow.
What is the probability that sum of both numbers is less than $7$ ?

Answer 1

Let’s figure out the number of possible outcomes. Each has $6$ possible outcomes $(1,2,3,4,5,6)$, so the combination of these two becomes $6\times 6=36$.

Below the first number in the parenthesis is the outcome of the first die and the second number is the outcome of the second die.

Now, of these 36 possible combinations, how many pair shows the sum of numbers is less than 7 of them:

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

Total possible outcomes are $15$

$Probability=\frac{desiredoutcomes}{totaloutcomes}$

Probability = $\frac{15}{36}=\frac{5}{12}$

So, the probability of the given question is $\frac{5}{12}$ .