Question

A game is played with 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 dice 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 find the value of $p+r-q$.

Answer 1

Answer: 7

Now, $2^{\text{nd }}$ blue result occur on or before the $10^{\text{th }}$ roll is equivalent to the occurrence of blue face atleast twice is 10 rolls
$n=10,p=\frac{1}{3},q=\frac{2}{3}$
$p(r\ge 2)=1-(P(0)+p(1))=1-\left({\left(\frac{2}{3}\right)}^{10}+10\cdot \frac{1}{3}{\left(\frac{2}{3}\right)}^9\right)$
$1-\frac{2^9\cdot 4}{3^9}=1-\frac{2^{11}}{3^9}=\frac{3^9-2^{11}}{3^9}$
$p=9,q=11,r=9$
$p+r-q=9+9-11=7$