# Marginal and Conditional probability

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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#### Solution Summary

The answer give the step by step procedure for the calculation of different basic probability problems. The problems are related to marginal probability, conditional probability and independence of events. Calculation of probability using normal distribution is also given.

Marginal & Conditional Probability for Marketing Survey

2. For a marketing survey, 200 customers were classified according to their age (in years) and their favorite type of donut.

Age of Customer Glazed Chocolate-covered Crème-filled Cake

<21 3 25 10 7

21-30 5 23 26 10

31-45 15 12 3 20

>45 29 5 1 6

(a) What is the probability that a person prefers Creme-filled donut and is age 30 years or less?

(b) What is the probability that either a person's favorite donut is glazed or that the person is less than 21 year of age?

(c) What is the probability that a person is between 21 and 30 years of age if that person favors chocolate-covered donuts?

(d) Are age and favorite donut independent variables? Why or why not?

Please see attached problem.

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