Question

A game is played with a special fair cubic die which has one red side, two blue sides, and three green sides. The result is the colour of the top side after the die has been rolled. If the die is rolled repeatedly, the probability that the second blue result occurs on or before the tenth roll, can be expressed in the form $\frac{3^p-2^q}{3^r}$ where p, q, r are positive integers, If $p^2+q^2+r^2.=280+x$. Find x

• A. 1
• B. 3
• C. 5
• D. 6

Answer 1

The correct option is B

3

B $\Rightarrow$ one red, 2 blue, 3 green

P(second blue result occurs on or before the tenth)

= 1 - P(second blue result occurs after ${10}^{th}$)

$=1-\left[{\left(\frac{4}{6}\right)}^{10}{+}^{10}{C}_1{\left(\frac{4}{6}\right)}^9\left(\frac{2}{6}\right)\right]$

$=\frac{6^{10}-4^{10}-4^{10}\times 5}{6^{10}}$

$=\frac{3^9-2^{11}}{3^9}$

$\Rightarrow p=9,q=11$ and $r=9$

$\Rightarrow p^2+q^2+r^2=9^2+{11}^2+9^2=81+121+81=283=280+3$

$\Rightarrow x=3$