A large corporation organized a ballot for all its workers...

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. a. What is the probability that a randomly chosen employee is a woman in favor of the plan? b. What is the probability that a randomly chosen employee is either a woman or a night-shift worker (or both)? c. Is employee gender independent of whether the night shift is worked? d. What is the probability that a female employee is a night-shift worker? 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?

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.
a. What is the probability that a randomly chosen employee is a woman in favor of the plan?
b. What is the probability that a randomly chosen employee is either a woman or a night-shift worker (or both)?
c. Is employee gender independent of whether the night shift is worked?
d. What is the probability that a female employee is a night-shift worker?
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?

Key Concepts

Complementary Probability

Complementary probability is the measure of the likelihood that a given event does not occur. It is obtained by subtracting the probability of the event from one, and is useful for computing the probabilities of events such as not favoring the bonus plan or not working the night shift.

Marginal Probability

Marginal probability represents the likelihood of a single event occurring without consideration of other variables. It involves summing over all possibilities of the other variables and is crucial when establishing the base rates, such as the overall proportion of night-shift workers or female employees.

Independence

Independence between two events implies that the occurrence of one event has no impact on the probability of the other. This concept is verified by checking whether the joint probability equals the product of the individual probabilities, which is key in determining if attributes like gender and work shift are related or independent.

Joint Probability

Joint probability is the likelihood of two events occurring simultaneously. In this context, it encompasses events like an employee being both a night-shift worker and a woman, or a woman who favors the bonus plan. It integrates multiple aspects of the scenario within a single probability measure.

Conditional Probability

This refers to the probability of one event occurring given that another event has occurred. It is important in scenarios where the outcome is affected by certain conditions, such as the gender or shift of an employee, and is calculated using the ratio of the joint probability of the two events to the probability of the given event.

Union of Events

The concept of the union of events deals with the probability that at least one of multiple events occurs. It is calculated by adding individual probabilities and then subtracting the probability of their intersection to avoid double counting, which is particularly useful when determining the probability of an employee belonging to at least one of several groups.

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