Question

A large corporation organized a ballot for all its workers on a new bonus plan. It was found that 65% of all night-shift workers favored the plan and that 40% of all female workers favored the plan. Also, 50% of all employees are night shift workers and 30% of all employees are women. Finally, 20% of all night-shift workers are women. question E) e. If 50% of all male employees favor the plan, what is the probability that a randomly chosen employee both does not work the night shift, and does not favor the plan.

Answer 1

Let N denote the event that an employees are night shift worker.

Similarly, let M and W be the event that employee is male or female respectively.

Given,

P(Favored | N) = 0.65

P(Favored | W) = 0.4

P(N) = 0.5

P(W) = 0.3

P(N and W) = 0.2

P(Favored | M) = 0.5

We need to find,

P(Does not Favor and No Night Shift)

P(No Night Shift) = 1 - P(N) = 1 - 0.5 = 0.5

P(M) = 1 - P(W) = 1 - 0.3 = 0.7

By law of total probability,

P(Favored) = P(Favored | M) P(M) + P(Favored | W) P(W)

= 0.5 * 0.7 + 0.4 * 0.3

= 0.47

P(Does not Favor) = 1 - P(Favored) = 1 - 0.47 = 0.53

P(Favored | N) = 0.65

P(Does not Favor | N) = 1 - 0.65 = 0.35

By Bayes theorem,

P(N | Does not Favor) = P(Does not Favor | N) * P(N) / P(Does not Favor)

= 0.35 * 0.5 / 0.53

= 0.3301887

P(No Night Shift | Does not Favor) = 1 - P(N | Does not Favor) = 1 - 0.3301887 = 0.6698113

By definition of conditional probability,

P(No Night Shift | Does not Favor) = P(Does not Favor and No Night Shift) / P(Does not Favor)

=> P(Does not Favor and No Night Shift) = P(No Night Shift | Does not Favor) * P(Does not Favor)

= 0.6698113 * 0.53

= 0.355