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# Knowledge of Permutations, Combinations, & Bayes' Theorem

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BUSA 3101
Example of Probability Problems Involving a
Knowledge of Permutations, Combinations,
And Bayes' Theorem

1. Dr. Deis has been teaching basic statistics for many years. He knows that 80 percent of the students will complete the assigned problems. He has also determined that among those who do their assignments, 90 percent will pass the course. Among those students who do not do their homework, 60 percent will pass. Daniel Kim took statistics last semester from Dr. Deis and received a passing grade. (In fact, he earned an excellent grade.) What is the probability he completed the assignments?

2. The credit department of Wal-Mart in Lake City, Georgia reported that 30 percent of their sales are cash, 30 percent are paid for by check at the time of the purchase, and 40 percent are charged. Twenty percent of the cash purchases, 90 percent of the checks, and 60 percent of the charges are for more than \$50. Ms. Norma "B" just purchased a new dress that cost \$120. What is the probability that she paid cash?

3. The Bembry Plastics Company has four raw-material suppliers. The following table shows the proportion purchased from each supplier and the percent of material that is defective from the supplier.
Supplier Percent purchased Percent
defective
Roberts,Inc.
Asmus Mfg.
Lewis, Ltd.
Melvin, Inc. 30.0
20.0
25.0
25.0 2.50
1.75
3.00
1.00

The material used a recent morning was defective. What is the probability it was purchased from Asmus Mfg?

4. A pollster randomly selects 4 of 10 available people. How many different groups of 4 are possible?

5. A telephone number consists of seven digits, the first three representing the exchange. How many different numbers are possible within the 537 exchange?

6. An overnight express company must include five cities on its route. How many different routes are possible assuming that it does not matter which order the cities are included in the routing?

7. A representative of the Environment Protection Agency (EPA) wants to select samples from 10 different landfills. He has 15 landfills from which he can collect samples. How many different samples are possible?

8. A national pollster has developed 15 questions designed to rate the performance of the president of the United States. The pollster will select 10 of these questions. How many different arrangements are there for the order of the 10 selected questions?

9. Carver Marketing Research, Inc. specializes in providing assessments of the prospects for women's apparel shops in shopping malls. Mikala Carver, president, reports that she assesses the prospects as good, fair, or poor. Records from previous assessments show that 60 percent of the time the prospects were rated as good, 30 percent of the time fair, and 10 percent of the time poor. Of those rated good, 80 percent made a profit the first year; of those rated fair, 60 percent made a profit the first year; and of those rated poor, 20 percent made a profit the first year. Elder's Apparel was one of Carver's clients. Elder's Apparel made a profit last year. What is the probability that it was given an original rating of poor?

10. Two boxes of men's Arrow shirts were received from the factory. Box 1 contained 25 sport shirts and 15 dress shirts. Box 2 contained 30 sport shirts and 10 dress shirts. One of the boxes was selected at random, and a shirt was chosen at random from that box to be inspected. The shirt was a sport shirt. Given this information, what is the probability that the box the sport shirt came from is box 1?

11. Refer to Exercise 10. What is the probability that the sport shirt came from box 2?

12. Refer to Exercise 10. Suppose the shirt selected from the box was a dress shirt (instead of a sport shirt). What is the probability that the dress shirt came from box 1?

13. Refer to Exercise 10. Suppose the shirt selected at random from the box was a dress shirt (instead of a sport shirt). What is the probability the dress shirt came from box 2?

14. Consideration is being given to forming a Super Ten football conference. The top 10 football teams in the country, based on past records, would be members of the Super Ten conference. Each team would play every team in the conference during the season. The team winning the most games would be declared the national champion. How many games would the conference commissioner have to schedule each year? (Remember, Oklahoma versus Michigan is the same as Michigan versus Oklahoma.)

#### Solution Preview

BUSA 3101
Example of Probability Problems Involving a
Knowledge of Permutations, Combinations,
And Bayes' Theorem

1. Dr. Deis has been teaching basic statistics for many years. He knows that 80 percent of the students will complete the assigned problems. He has also determined that among those who do their assignments, 90 percent will pass the course. Among those students who do not do their homework, 60 percent will pass. Daniel Kim took statistics last semester from Dr. Deis and received a passing grade. (In fact, he earned an excellent grade.) What is the probability he completed the assignments?

Let we assume that P(A) denote the probability of those student who will completed their assignments. i.e. P(A) = 0.8
P(B) = the probability of those student who will completed their assignments
i.e. P(B) = 0.2

Now Let P(E|A) denote the those student who do their assignments, and passed i.e. P(E|A) = 0.9
P(E|B) denote the those student who do their assignments, and passed i.e. P(E|B) = 0.6

Then From Baye's theorem

 Daniel Kim took statistics last semester from Dr. Deis and received a passing grade. Then probability of completing his grade

= (0.9*0.8)/(0.9*0.8 + 0.6*0.2)
= 0.72/0.84 = 6/7 = 0.86 (approx.)

2. The credit department of Wal-Mart in Lake City, Georgia reported that 30 percent of their sales are cash, 30 percent are paid for by check at the time of the purchase, and 40 percent are charged. Twenty percent of the cash purchases, 90 percent of the checks, and 60 percent of the charges are for more than \$50. Ms. Norma "B" just purchased a new dress that cost \$120. What is the probability that she paid cash?

There are three method of sales by cash (A), by check(B) and by charge (C). then according to the problem, we can see that

P(A) = 0.3 , P(B) = 0.3 and P(C) = 0.4

Also given that, probability of purchasing more than \$ 50 is,by cash P(E|A) = 0.4, by check P(E|B) = 0.9 and by charge P(E|C) = 0.6
Ms. Norma purchased a new dress that cost \$120 (clearly more than \$50). We have to find the probability if she paid cash i.e. P(A|E)
Then by Baye's theorem

...

#### Solution Summary

The solution examines knowledge of permutations, combinations, and Bayes' Theorem.

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