JEE Main 2022 (Online) 27th July Evening ShiftQuestion 1
A six faced die is biased such that $3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1)$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is :
Probability
Solution
Let $3P(\text{each prime})=6P(\text{each composite})=2P(1)=t$: each prime $\frac t3$ (faces $2,3,5$), each composite $\frac t6$ (faces $4,6$), $P(1)=\frac t2$. Total $t+\frac t3+\frac t2=\frac{11t}6=1\Rightarrow t=\frac6{11}$. Perfect squares on the die: $1,4$: $p=\frac t2+\frac t6=\frac3{11}+\frac1{11}=\frac4{11}$. Mean of $X$ in two throws $=2p=\frac8{11}$. Answer: (D).