A school has 250 employees categorized by task and gender in...

A school has 250 employees categorized by task and gender in the following table. $$\begin{array}{|l|c|c|c|}\hline & \text { Teaching } & \text { Administrative } & \text { Support } \\\hline \text { Male } & 84 & 14 & 52 \\\hline \text { Female } & 56 & 26 & 18 \\\hline\end{array}$$ An employee is randomly selected. Let $A$ be the event that he/she is an administrative staff member, T teaching staff, S support, $M$ male, and F female. a) Write down the following probabilities: $P(F), P(F \cap T), P\left(F \cup A^{\prime}\right), P\left(F^{\prime} | A\right)$ b) Which events are independent of $F$, which are mutually exclusive to $F$. Justify your choices. c) (i) Given that $90 \%$ of teachers, as well as $80 \%$ of the administrative staff and $30 \%$ of the support staff, own cars, find the probability that a staff member chosen at random owns a car. (ii) Knowing that the randomly chosen staff member owns a car, find the probability that he/she is a teacher.

A school has 250 employees categorized by task and gender in the following table.
$$\begin{array}{|l|c|c|c|}\hline & \text { Teaching } & \text { Administrative } & \text { Support } \\\hline \text { Male } & 84 & 14 & 52 \\\hline \text { Female } & 56 & 26 & 18 \\\hline\end{array}$$
An employee is randomly selected. Let $A$ be the event that he/she is an administrative staff member, T teaching staff, S support, $M$ male, and F female.
a) Write down the following probabilities: $P(F), P(F \cap T), P\left(F \cup A^{\prime}\right), P\left(F^{\prime} | A\right)$
b) Which events are independent of $F$, which are mutually exclusive to $F$. Justify your choices.
c) (i) Given that $90 \%$ of teachers, as well as $80 \%$ of the administrative staff and $30 \%$ of the support staff, own cars, find the probability that a staff member chosen at random owns a car.
(ii) Knowing that the randomly chosen staff member owns a car, find the probability that he/she is a teacher.

Key Concepts

Law of Total Probability

This law allows the calculation of an overall probability by partitioning the sample space into mutually exclusive parts and summing the probabilities over these parts. It is especially useful when dealing with probabilities that span different categories or groups.

Bayes' Theorem

Bayes' Theorem provides a framework to update the probability of an event based on new evidence. It is particularly important in reversing conditional probabilities, such as determining the probability of belonging to a certain category given a known outcome.

Mutually Exclusive Events

This concept refers to two events that cannot occur at the same time; their probability of occurring together is zero. It is used to identify combinations of outcomes that are impossible, aiding in the analysis of scenarios where events are disjoint.

Event Operations (Union, Intersection, Complement)

These are fundamental operations in set theory and probability. The union represents the occurrence of at least one of two events, the intersection represents the occurrence of both events simultaneously, and the complement refers to the non-occurrence of an event. These operations enable the calculation of combined or mutually related events.

Conditional Probability

This concept deals with the probability of an event occurring given that another event has already occurred. It is central for problems where outcomes depend on a known condition, such as determining the likelihood of a gender category within a specific staff category or reversing conditional probabilities.

Independence of Events

Two events are independent if the occurrence of one does not affect the probability of the other. Recognizing independent events helps in simplifying probability calculations, especially in cases where separate characteristics are evaluated without influence on each other.

Probability Basics

This is the foundation for understanding random experiments, events, and outcomes. It involves defining a sample space and calculating probabilities as ratios of favorable outcomes to the total number of outcomes, which is crucial for working with any categorical data or contingency tables.