A manager has available a pool of 8 employees who could be...

A manager has available a pool of 8 employees who could be assigned to a project-monitoring task. 4 of the employees are women and 4 are men. 2 of the men are brothers. The manager is to make the assignment at random so that each of the 8 employees is equally likely to be chosen. Let $A$ be the event "chosen employee is a man" and $B$ the event "chosen employee is one of the brothers." a. Find the probability of $A$. b. Find the probability of $B$. c. Find the probability of the intersection of $A$ and $B$.

A manager has available a pool of 8 employees who could be assigned to a project-monitoring task. 4 of the employees are women and 4 are men. 2 of the men are brothers. The manager is to make the assignment at random so that each of the 8 employees is equally likely to be chosen. Let $A$ be the event "chosen employee is a man" and $B$ the event "chosen employee is one of the brothers."
a. Find the probability of $A$.
b. Find the probability of $B$.
c. Find the probability of the intersection of $A$ and $B$.

Key Concepts

Probability of an Event

The probability of an event is the measure of how likely that event is to occur, defined as the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This basic concept underpins many probability problems and is calculated when each outcome is equally likely.

Intersection of Events

The intersection of events encompasses outcomes that satisfy all given events simultaneously. In probability, calculating the intersection involves identifying the common outcomes between events and determining their likelihood relative to the total sample space. This is useful for understanding scenarios where multiple conditions must be met at once.

Uniform Random Selection

Uniform random selection refers to the process where every outcome in the sample space is equally likely to occur. In probability problems, it means that when selecting an element from a set, each element has the same chance of being chosen, which simplifies the calculation to counting favorable outcomes over the total number of outcomes.

Sample Space

The sample space is the complete set of all possible outcomes in a probability experiment. In the context of random selection, it includes every potential candidate that could be chosen, and knowing the size of the sample space is essential for calculating probabilities by establishing the denominator in the ratio of favorable outcomes to total outcomes.

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