A producer assigns three roles to 20 actors. Roles were younger boy, old man and disable man. Find probability that 2 or more actors perform old man in the film.
P(X > 2) = ?
P(X > 2) = 1 - [P(X = 1) + P(X = 0)]
P(X > 2) = 1 - P(X = 1) - P(X = 0)
Here , we have to use the formula `P(X = x) = (nCx) p^x q^(n-x)
Here, x = 2, n = 20, q = 1 - p
What would be the value of p here ?`
As we do not have any other information about the distribution of the roles, we can assume they are distributed uniformly. Unless we consider roles in the movies to know more about the distribution. Hence, We have: $$\mathbb{P}(X=2) = \frac{\binom{20}{2}2^{18}}{3^{20}} = \binom{20}{2} \left(\frac{2}{3}\right)^{18}\left(\frac{1}{3}\right)^{2}$$ $$\mathbb{P}(X=1) = \frac{\binom{20}{1}2^{19}}{3^{20}}=\binom{20}{1} \left(\frac{2}{3}\right)^{19}\left(\frac{1}{3}\right)^{1}$$ $$\mathbb{P}(X=0) = \frac{2^{20}}{3^{20}} = \binom{20}{0} \left(\frac{2}{3}\right)^{20}\left(\frac{1}{3}\right)^{0}$$ In the above, we choose actors for the old man role, and the others take the other two roles respectively. Now: $$\mathbb{P}(X>2) = 1 - \left(\mathbb{P}(X = 0) + \mathbb{P}(X = 1) + \mathbb{P}(X=2)\right)$$