Correct Answer - Option 4 :
\(\dfrac{25}{216}\)
Concept:
For the binomial distribution of probability
Where n is the total number of cases, p is probability of favorable cases and q is probability of unfavorable cases(1 - p)
Calculation:
Given mean = np = \(2\over3\) and variance = npq = \(5\over9\)
\(\rm variance \over mean\) = \({5\over9}\over{2\over3}\)
q = \(5\over6\)
p = 1 - q = \(1\over6\)
np = \(2\over3\)
n = 4
∴ The probability that random variable D = 2
P = \(\rm {^nC_2}p^2q^{n-2}\)
P = \(\rm {^4C_2}\times\left({1\over6}\right)^2\times\left({5\over6}\right)^2\)
P = \(\rm 6\times{1\over36}\times{25\over36}\)
P = \(25\over216\)