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Marginal & Conditional Probabilities

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A box contains six parts. Two of the parts are defective and four are ok. If three of the six parts are selected from the bin, how large is the sample space? Which counting rule did you use, and why? For this sample space, what is the probability that exactly one of the three sampled parts is defective?

A business has 20 employees. Six of these employees will be selected randomly to be interviewed as part of an employee satisfaction program. How many different groups of six can be selected?

Alex, Alicia, and Juan fill orders in a fast-food restaurant. Alex incorrectly fills 20% of the orders he takes. Alicia incorrectly fills 12% of the orders she takes. Juan incorrectly fills 5% of the orders he takes. Alex fills 30% of all orders, Alicia fills 45% of all orders, and Juan fills 25% of all orders. An order has just been filled.

a. What is the probability that Alicia filled the order?

b. If the order was filled by Juan, what is the probability that it was filled correctly?

c. Who filled the order is unknown, but the order was filled incorrectly. What are the
revised probabilities that Alex, Alicia, or Juan filled the order?

d. Who filled the order is unknown, but the order was filled correctly. What are the revised
probabilities that Alex, Alicia, or Juan filled the order?

Purchasing Survey asked purchasing professionals what sales traits impressed them most in a sales representative. Seventy-eight percent selected "thoroughness." Forty percent responded "knowledge of your own product." The purchasing professionals were allowed to list more than one trait. Suppose 27% of the purchasing professionals listed both "thoroughness" and "knowledge of your own product" as sales traits that impressed them most. A purchasing professional is randomly sampled.
a. What is the probability that the professional selected "thoroughness" or "knowledge of your own product"?
b. What is the probability that the professional selected neither "thoroughness" nor "knowledge of your own product"?
c. If it is known that the professional selected "thoroughness," what is the probability that the professional selected "knowledge of your own product"?
d. What is the probability that the professional did not select "thoroughness" and did select "knowledge of your own product"?

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Solution Summary

The solution provides step by step method for the calculation of probabilities and conditional probabilities. Formula for the calculation and Interpretations of the results are also included.

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Normal Probability based on Z score

A soft drink bottling company maintains records concerning the number of unacceptable bottles of soft drink obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.035. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II.
a. Give an example of a simple event.
b. Give an example of a joint event.
c. If a filled bottle of soft drink is selected at random, what is the probability that it is a nonconforming bottle?
d. If a filled bottle of soft drink is selected at random, what is the probability that it was filled on machine II?
e. If a filled bottle of soft drink is selected at random, what is the probability that it was filled on machine I and is a conforming bottle?

3. According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 14.4%, and the standard deviation of the annual return was 20.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 7.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.
a. Find the probability that the return for common stocks will be greater than 0%.
b. Find the probability that the return for common stocks will be less than 0%.
c. Find the probability that the return for common stocks will be less than 15%.
d. Find the probability that the return for common stocks will be greater than 20%.
e. Find the probability that the return for common stocks will be greater than 30%.
f. Find the probability that the return for common stocks will be less than -10%.
Hint: There are many ways to attack this third problem in the HW. If you would like the normal distribution table so you can draw the pictures

See attached file for full problem description.

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