By the end of this chapter, the student should be able to:

Understand and use the terminology of probability.

Determine whether two events are mutually exclusive and whether two events are independent.

Calculate probabilities using the Addition Rules and Multiplication Rules.

Construct and interpret Contingency Tables.

Construct and interpret Venn Diagrams.

Construct and interpret Tree Diagrams.

It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs.

You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach.

Collaborative Exercise

Your instructor will survey your class. Count the number of students in the class today.

Raise your hand if you have any change in your pocket or purse. Record the number of raised hands.

Raise your hand if you rode a bus within the past month. Record the number of raised hands.

Raise your hand if you answered "yes" to BOTH of the first two questions. Record the number of raised hands.

Use the class data as estimates of the following probabilities. \(P(\text{change})\) means the probability that a randomly chosen person in your class has change in his/her pocket or purse. \(P(\text{bus})\) means the probability that a randomly chosen person in your class rode a bus within the last month and so on. Discuss your answers.

Find \(P(\text{change})\).

Find \(P(\text{bus})\).

Find \(P(\text{change AND bus})\). Find the probability that a randomly chosen student in your class has change in his/her pocket or purse and rode a bus within the last month.

Find \(P(\text{change|bus})\). Find the probability that a randomly chosen student has change given that he or she rode a bus within the last month. Count all the students that rode a bus. From the group of students who rode a bus, count those who have change. The probability is equal to those who have change and rode a bus divided by those who rode a bus.