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Q.

A six face die is biased such that $3 \times P$ (a prime number) $= 6 \times P$ (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 110 times of the mean of X is:

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Detailed Solution

Let $\frac{P (a p r i m e n u m b e r)}{2} = \frac{P (a c o m p o s i t e)}{1} = \frac{P (1)}{3} = k$
So, $P$ (a prime number) $= 2 k, P$ (a composite number) $= k$ $& P (1) = 3 k$
$& 3 \times 2 k + 2 \times k + 3 k = 1 \Rightarrow k = \frac{1}{11}$
$P$ (success) $= P$ $(1 o r 4) = 3 k + k = \frac{4}{11}$
Number of trials, $n = 2$
Therefore, mean $= n p = 2 \times \frac{4}{11} = \frac{8}{11}$

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