Question
A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is
A.
$$\frac{8}{{729}}$$
B.
$$\frac{8}{{243}}$$
C.
$$\frac{1}{{729}}$$
D.
$$\frac{8}{{9}}$$
Answer :
$$\frac{8}{{243}}$$
Solution :
A pair of fair dice is thrown, the sample space $$S$$ = (1, 1) (1,2) (1, 3) . . . . = 36
Possibility of getting 9 are (5,4) , (4, 5), (6,3), (3,6)
∴ Probability of getting score 9 in a single throw
$$\eqalign{
& = \frac{4}{{36}} \cr
& = \frac{1}{9} \cr} $$
∴ Probability of getting score 9 exactly twice
$$\eqalign{
& = {\,^3}{C_2} \times {\left( {\frac{1}{9}} \right)^2}.\left( {1 - \frac{1}{9}} \right) \cr
& = \frac{{3!}}{{2!}} \times \frac{1}{9} \times \frac{1}{9} \times \frac{8}{9} \cr
& = \frac{{3.2!}}{{2!}} \times \frac{1}{9} \times \frac{1}{9} \times \frac{8}{9} \cr
& = \frac{8}{{243}} \cr} $$