A die is thrown n times. A random variable X denotes the number of times, the number on the dice is greater than 4 and \(\mathrm{P}(\mathrm{X} = 1) = 2\mathrm{P}(\mathrm{X} = 2)\). The value of n is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
A number greater than 4 means 5 or 6, so in each throw
\( \displaystyle p = \frac{2}{6} = \frac{1}{3}, \quad q = \frac{2}{3} \)
The throws are independent, so X is binomial, \(P(X = r) = {}^{n}C_{r}\,p^{r}q^{n-r}\). The given condition is
\({}^{n}C_{1}\,p\,q^{n-1} = 2 \times {}^{n}C_{2}\,p^{2}q^{n-2}\)
\( \displaystyle n\,p\,q^{n-1} = 2 \times \frac{n(n-1)}{2}\,p^{2}q^{n-2} \)
Cancelling \(n\,p\,q^{n-2}\):
\(q = (n-1)p\)
\( \displaystyle \frac{2}{3} = \frac{n-1}{3} \)
\(n = 3\)