# Marginal & Conditional Probabilities

Section 3.1: Basic Concepts of Probability and Counting

1. License plates are made using 2 letters followed by 3 digits. How many different plates can be made if repetition of letters and digits is allowed? (References: example 4 page 135, end of section exercises 13 - 16 page 142 and 35 - 36 page 144)

2. In 2005 the stock market took some big swings up and down. One thousand investors were asked how often they tracked their investments. The table below shows their responses. What is the probability that an investor tracks the portfolio monthly? (References: example 6 page 137, end of section exercises 45 - 48 page 145)

How often tracked? Response

Daily 235

Weekly 278

Monthly 292

Few times a year 136

Do not track 59

Section 3.2: Conditional Probability and the Multiplication Rule

3a. In a battleground state, 40% of all voters are Republicans. Assuming that there are only two parties - Democrat and Republican, if two voters are randomly selected for a telephone survey, what is the probability that they are both Republicans? Round your answer to 4 decimal places (References: example 4 page 152, end of section exercises 19 - 21 page 156 - 157)

b. You are dealt 2 cards from a shuffled deck of 52 cards, without replacement. There are four suits of 13 cards each in a deck of cards; two of them are black and two of them are red. What is the probability that both cards are black? Round your answer to 3 decimal places (References: example 3 page 151.

4. The table below shows the drink preferences for people in 3 different age groups. If one of the 255 subjects is randomly chosen, what is the probability that the person drinks cola given they are over 40? Round your answer to 3 decimal places. (References: example 4 and 5 page 152 - 153, end of section exercises 15, 16, 23 , 24 page 156)

Water Orange juice Cola

Under 21 years 40 25 20

21 - 40 years 35 20 30

Over 40 years 20 30 35

Section 3.3: The Addition Rule

5. a. The table below shows the drinking habits of adult men and women.

Non-Drinker Occasional Drinker Regular Drinker Heavy Drinker Total

Men 387 45 90 37 559

Women 421 46 69 34 570

Total 808 91 159 71 1,129

If one of the 1,129 people is randomly chosen, what is the probability that the person is a man or a non-drinker? Round your answer to 3 decimal places. (References: example 4 page 163, end of section exercises 23 - 26 page 168 - 169)

b. The table show drinking habits of adult men and women.

Non-Drinker Occasional Drinker Regular Drinker Heavy Drinker Total

Men 387 45 90 37 559

Women 421 46 69 34 570

Total 808 91 159 71 1,129

If one of the 1,129 people is randomly chosen, what is the probability that the person is a non-drinker or a heavy drinker? Round your answer to 3 decimal places. (References: example 4 page 163, end of section exercises 23 - 26 page 168 - 169)

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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.

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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