Question

A game is played with 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 is rolled. If the die is rolled repeatedly, the probability that second blue result occurs on or before the tenth roll can be expressed in the form of $\frac{3^p-2^q}{3^r}$ where $p,q,r$ are positive integers. Then the value of $p+r-q$ is

Answer 1

$2^{\text{nd}}$ blue result occurs on or before ${10}^{\text{th}}$ roll is equivalent to the occurrence of blue face at least twice in $10$ rolls.
$n=10,P\left(\text{blue}\right)=\frac{1}{3},P\left(\text{not blue}\right)=\frac{2}{3}$

$P\left(\text{blue does not occur in first ten rolls}\right)={}^{10}{C}_0{\left(\frac{1}{3}\right)}^0{\left(\frac{2}{3}\right)}^{10}$
$P\left(\text{blue occurs only one time in first ten rolls}\right)={}^{10}{C}_1{\left(\frac{1}{3}\right)}^1{\left(\frac{2}{3}\right)}^9$

$P\left(2^{\text{nd}}\text{blue occurs on or before tenth roll}\right)=1-[{\left(\frac{2}{3}\right)}^{10}+10\left(\frac{1}{3}\right){\left(\frac{2}{3}\right)}^9]=\frac{3^9-2^{11}}{3^9}$
$p=9,q=11,r=9$
$p+r-q=9+9-11=7$