The faculty of the mathematics department at Joliet Junior...

The faculty of the mathematics department at Joliet Junior College is composed of 4 females and 9 males. Of the 4 females, 2 are under age $40,$ and of the 9 males, 3 are under age $40 .$ Find the probability that a randomly selected faculty member is: (a) Female or under age 40 (b) Male or over age 40

The faculty of the mathematics department at Joliet Junior College is composed of 4 females and 9 males. Of the 4 females, 2 are under age $40,$ and of the 9 males, 3 are under age $40 .$ Find the probability that a randomly selected faculty member is:
(a) Female or under age 40
(b) Male or over age 40

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Key Concepts

Probability

Probability is the measure of the chance that a particular event will occur, calculated as the number of favorable outcomes divided by the total number of possible outcomes in the sample space. This foundational concept underpins problems where one must determine the likelihood of various events.

Counting Techniques

Counting techniques involve methods for determining the number of ways events can occur, such as using direct counts or combinatorial methods. They are essential in probability to accurately enumerate both the total number of outcomes and the number of favorable outcomes.

Set Operations in Probability

Set operations in probability, such as unions and intersections, allow us to combine events. The union of events represents outcomes that belong to at least one event, while the intersection represents outcomes common to multiple events. Understanding these operations is key when dealing with compound events.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a method to correctly calculate the probability of the union of two events by adding the probabilities of each event and then subtracting the probability of their intersection, to avoid double counting. This principle is crucial when events overlap and are not mutually exclusive.

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