Section 5.3 Completed Notes

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Conditional ProbabilityThe probability of an event occurring GIVEN we know that

some other event has occurred.

FOR EXAMPLE: Consider rolling a 2 on a die. This event has a probability of 1/6. Now suppose we know that we rolled an even number. This changes the probability of rolling a 2 to 1/3.

Sample Space for rolling a die Sample Space for rolling a die GIVEN it is an

even number.

We can write the probability of rolling a 2 on a die GIVEN we know that we rolled an even number as:

P(rolling a 2 | rolling an even number)

Section 5.3 Completed Notes

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A study of the students taking distance learning courses at a university finds that they are mostly older students not living in the university town. Choose a distance-learning student at random. Let A be the event that the student is 25 years old or older and B the event that the student is local. The study finds that P(A) = 0.7, P(B) = 0.25, and P(A and B) = 0.05. What is the conditional probability that a student is local given that he or she is less than 25 years old?

Section 5.3 Completed Notes

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Here are the counts (in thousands) of earned degrees in the United States in the 2001-2002 academic year, classified by level and by the sex of the degree recipient:

(a) If you choose a degree recipient at random, what is the probability that the person you choose is a woman?

(b) What is the conditional probability that you choose a woman, given that the person chosen received a professional degree?

(c) What is the probability that you choose a doctorate degree recipient, given that the person chosen is a man?

Here are the counts (in thousands) of the top four most popular careers in the United States by job name and by the sex of the employee:

(a) What is the probability that a randomly chosen employee is a woman?

(b) What is the probability that a Network Systems analyst is a woman?

(c) What is the probability of selecting a female Network Systems Analyst?

Section 5.3 Completed Notes

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Conditional Probabilities lead us to the General Multiplication Rule:

General Multiplication Rule for ANY Two Events

The probability that both of two events A and B happen together can be found by

P(A and B) = P(A)P(B|A)

Suppose that 10% of adults belong to health clubs, and 40% of these health club members go the club at least twice a week. Of those who don't belong, 73% go to the health club less than 2 times per week. What percent of all adults go to a health club at least twice a week?

Section 5.3 Completed Notes

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Using tree diagrams...Suppose that 10% of adults belong to health clubs, and 40% of these health club members go the club at least twice a week. Of those who don't belong, 73% go to the health club less than 2 times per week. What percent of all adults go to a health club at least twice a week?

Drivers have a wide selection of fuel to purchase for use in their cars. A recent study found that 48% of all drivers use Regular unleaded fuel, 29% use Unleaded Plus, and 23% use Premium unleaded. Of those that use regular fuel 86% spend less than $35 on a tank of gas. However, of those that use Unleaded Plus, 62% spend more than $35 on a tank of gas and for those that use Premium Unleaded this percentage rises to 93%. What is the probability that a randomly chosen driver spends less than $35 on a tank of gas?

Section 5.3 Completed Notes

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(a) What's the probability of spending more than $35 on a tank of gas and using regular unleaded fuel?

(b) What's the probability of spending more than $35 on a tank of gas?

(c) What's the probability of spending less than $35 on a tank of gas and using premium unleaded fuel?

(d) What's the probability of using plus unleaded fuel given you have spent more than $35 on a tank of gas?

Call a person educated if he or she holds at least a bachelor's degree. Call a person who earns at least $100,000 a year prosperous. The Census Bureau says that in 2004, 28% of American adults (age 25 or older) were educated. Among these educated adults 13% were prosperous. What percent of adults were both educated and prosperous?

Section 5.3 Completed Notes

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Independence

We can also re-think how we define whether two events are independent or not.

Remember we said events A and B are independent if knowing that one has occurred - let's say A - doesn't change the probability that the other - B - occurs.

Can we write this as a statement about probabilities?

Homework:p.329#s63-69odd,73,77,79,83,85,87,91-99odd,104,105,106