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Probability Measures & Calculation: Examples and Exercises

1. Valid measures of probability

The values below, which ones could be valid measures of probability?

1. 0
2. 15
3. -0.74
4. .09

1.What does it mean if an event has probability ?

2. A coin is flipped three times ? What is the sample space?

3. What is the probability of getting exactly two heads?

4. In a certain game, the probability that the first player will win is 0.96. What does this mean using the frequentist interpretation of probability?

5. A college student is selected at random. Let event A be the event that the student’s age is less than 21. Let B be the event that the student’s age is at least 18. Let C be the event that the students is between 20 and 25, inclusive. Describe the following events in words, as simply as possible.

1. (A or C)
2. (not B)
3. (A and B)

6. The number of hours students took to finish a project are given below.

 Hours Number of students 4 2 5 12 6 16 7 11 8 12 9 18 10 1

A student is selected at random. Let A be the event the student took less than 6 hours. Let B be the event the student took between 5 and 7 hours inclusive. How many outcomes are in (A and B)?

7. A committee is made up of five people: Amanda, Brad, Cathy, Dave, and Emily. Three people are selected to serve on a subcommittee. Let A be the event that both Brad and Dave are selected. Let B be the event that at least one woman is chosen. Are A and B mutually exclusive? Explain.

8. The following frequency distribution analyzes the scores on a math test. Find the probability that a score under 76 was achieved.

 Score Frequency 40-59 13 60-75 5 76-82 9 83-94 10 95-100 15

9. The following gives the relative frequency for executives at a particular company. An executive is selected at random. Find the probability that the selected executive has a salary of at least \$60,000.

 Salary Relative Frequency Under \$60,000 0.06 \$60,000 – under \$70,000 0.37 \$70,000 – under \$80,000 0.28 \$80,000 – under \$90,000 0.06 \$90,000 – under \$100,000 0.14 \$100,000 or more 0.09
1. A card is drawn from a standard 52-card deck.
2.  What is the probability that the card is a face card (K, Q, or J)?
3. What is the probability that the card is a 5?
4. Use the Special Addition Rule to find the probability the card is a face card or a 5.

For a particular experiment and events A and B, P(A) = 0.29, P(B) = 0.23, and P(A and B) = 0.15. Find P(A or B).

10. A big department store employs a total of 180 people. Of these, 6.7% are managers, 19.1% are floor assistants, and the rest are salespeople. If we choose an employee at random, find the probability the employee is not a manager.

11. The table below lists the blood type and gender of patients in a hospital. One patient is selected at random.

 O A B AB Total Female 91 45 55 3 194 Male 89 56 45 14 204 Total 180 101 100 17 398

1.What is the probability that the person selected is a male with type 0 blood?

2. What is the probability that the person has type B blood, given that they are female?

3. If the selected person has type AB blood, what is the probability they are male?

4. What is the probability the selected person is female or has type AB blood?

12. You are dealt two cards at random from a standard deck. What is the probability the first card is a king and the second is a queen?

13. If P(A) = 0.76, P(B) = 0.88, and P(A and B) = 0.6688, are A and B independent events?Show.

14. The following table compare cause of death for smokers and nonsmokers from a sample of 1000 people.

 Cancer Heart Disease Other Total Smoker 230 233 65 528 Nonsmoker 136 199 137 472 Total 366 432 202 1000

A. If a person is selected at random, find the probability the individual died of cancer.

B. Find the probability the person died of cancer, given they were a smoker.

C. Are the events “smoker” and “died of cancer” independent? Explain how you know this

D. A bag contains 18 red chips and 9 blue chips. Two chips are selected at random without replacement. Draw a tree diagram giving the possible outcomes and their probabilities.

15. A bag contains 13 red chips and 12 blue chips. Two chips are selected at random without replacement. What is the probability the two chips are the same color?

Extra credit: Two cards are chosen at random from a standard 52-card deck. Without replacement. What is the probability the first is an ace and the second is a diamond? Show work. Write the answer as a reduced fraction or round to four places after the decimal.