JEE MAIN - Mathematics (2022 - 27th July Evening Shift - No. 12)
A six faced die is biased such that
$$3 \times \mathrm{P}($$a prime number$$)\,=6 \times \mathrm{P}($$a composite number$$)\,=2 \times \mathrm{P}(1)$$.
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 :
Explanation
Let P(a prime number) = $$\alpha$$
P(a composite number) = $$\beta$$
and P(1) = $$\gamma$$
$$\because$$ $$3\alpha = 6\beta = 2\gamma = k$$ (say)
and $$3\alpha + 2\beta + \gamma = 1$$
$$ \Rightarrow k + {k \over 3} + {k \over 2} = 1 \Rightarrow k = {6 \over {11}}$$
Mean = np where n = 2
and p = probability of getting perfect square
$$ = P(1) + P(4) = {k \over 2} + {k \over 6} = {4 \over {11}}$$
So, mean $$ = 2\,.\,\left( {{4 \over {11}}} \right) = {8 \over {11}}$$
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