Question

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly $9$twice is

• A. $\frac{1}{729}$
• B. $\frac{8}{9}$
• C. $\frac{8}{729}$
• D. $\frac{8}{243}$

Answer 1

The correct option is D

$\frac{8}{243}$

Explanation for the correct option:

To find the probability:

When two dice are thrown, the total number of sample space is $36$

Possibility of getting a score of $9$in a single throw are $(3,6),(4,5),(5,4),(6,3)$

Thus the probability of getting a score of $9$ $=\frac{4}{36}=\frac{1}{9}$

and the probability of not getting a score of $9$ $=1-\frac{1}{9}=\frac{8}{9}$

Hence, the probability of getting a score of exactly $9$twice in $3$independently throws

$={}^{3}C_2{\left(\frac{1}{9}\right)}^2\left(\frac{8}{9}\right)$ (Binomial probability distribution)

$$\begin{gathered} =3\times \left(\frac{1}{81}\right)\times \left(\frac{8}{9}\right) \\ =\frac{8}{243} \end{gathered}$$

Hence the correct option is D