Question

If two dice are thrown simultaneously, then the probability that the sum of the numbers which come up on the dice to be more than $5$ is

• A. $\frac{5}{36}$
• B. $\frac{1}{6}$
• C. $\frac{5}{18}$
• D. $\frac{7}{18}$
• E. None of these

Answer 1

The correct option is E

None of these

Explanation for the correct answer:

When $n$ dice are rolled the number of outcomes in sample space are given as

$n\left(S\right)=6^n$

Two dice are rolled hence the value of $n=2$

$\Rightarrow$$n\left(S\right)=6^2=36$

Let $E$ be the event where the sum of numbers coming up on the dice is less than or equal to $5$

$\Rightarrow$$E=\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(3,1\right),\left(3,2\right),\left(4,1\right)\right\}$

$\Rightarrow$ $n\left(E\right)=10$

To obtain a sum of more than $5$ and obtain a sum of less than or equal to $5$ are mutually exclusive events.

The probability of sum of numbers being more than $5$$=1-$probability of sum of numbers being less than or equal to $5$

Let the probability of sum of numbers being more than $5$$=P\left(E>5\right)$

The probability of sum of numbers being less than or equal to $5$$=P\left(E\le 5\right)$

$$\begin{gathered} \therefore P\left(E>5\right)=1-P\left(E\le 5\right) \\ =1-\frac{n\left(E\right)}{n\left(S\right)} \\ =1-\frac{10}{36} \\ =\frac{26}{36} \\ =\frac{13}{18} \end{gathered}$$

Hence, option (E) is the correct answer