Question

A club with 23 members is voting on a proposal. A group of 5 has decided to vote in favor of the proposal while each of the rest of the members vote independently with a chance of 44% in favor of the proposal. What are the chances that the proposal will get the majority vote.

Answer 1

Of the 23 members, 5 will vote in favor of the proposal

From the remaining n = 18, each will vote in favor with probability, p = 0.44

q = 1 - p = 0.56

P(the proposal will get the majority vote) = P(number of total votes in favor 12)

= P(number of votes in favor from the remaining 18 7)

np = 18x0.44 = 7.92

nq = 18x0.56 = 10.08

Normal approximation for binomial can be used here since both np and nq 5

P(X < A) = P(Z < (A - mean)/standard deviation)

Mean = np = 7.92

Standard deviation =

=

= 2.106

P(the proposal will get majority vote) = P(X 7)

= 1 - P(X < 6.5) (with continuity correction)

= 1 - P(Z < (6.5 - 7.92)/2.016)

= 1 - P(Z < -0.67)

= 1 - 0.2502

= 0.7498

(Approximately 75%)