Ail investing club has n members The club has decided to vot

Ail investing club has n members. The club has decided to vote on whether or not to jointly invest in each of M different funds. For each fund, all n members vote. If no more than 1 member votes NO, then the club will invest in the fund. Investor preferences are modeled as independent from fund to fund with each investor voting to invest in a given fund with probability p. Investors are also assumed to vote independently of each other. What is the probability mass function for the number of funds in which the club invests?

Solution

The probability that no more than 1 will vote no is

P(no more than 1) = P(1 no) + P(0 no)

Using binomial distribution,

P(no more than 1) = n p^(n - 1) (1 - p) + p^n

Thus,

P(will invest) = n p^(n - 1) (1 - p) + p^n = P

Thus, the probability mass function for x, the number of investments, is

pmf(x) = C(n, x) P^x (1 - P)^{n - x}

where P = n p^{n - 1} (1 - p) + pⁿ. [ANSWER]