The production of a certain volatile commodity is the outcome of a stochastic production function given by \(Y\) \(=L^{5} K^{25} e^{V}\), where \(V\) is a random variable having the cumulative distribution function \(F(V)=\frac{1}{1+e^{-2(v-1)}}, L\) denotes units of labor and \(K\) denotes units of capital.

a. If labor is applied at 9 units and capital is applied at 16 units, what is the probability that output will exceed 12 units?

b. Given the input levels applied in (a), what is the probability that output will be between 12 and 16 units?

c. What level of capital and labor should be applied so that the probability of producing a positive profit is maximized when output price is \(\$ 10\), labor price is \(\$ 5\), and capital price is \(\$ 10\) ?

d. What is the value of the maximum probability of obtaining positive profit?