1.  Drinking and Driving

In many states a motorist is legally drunk or driving under the influence (DUI) if his or her blood alcohol concentration is .10% or higher. When a suspected DUI offender is pulled over, police often request a sobriety test called a breathalyzer, in which the suspected offender breathes into a machine that reports a blood alcohol level. Although the breathalyzers are remarkably precise, they do exhibit some measurement error. Because of that variability, the possibility exists that a driver's true blood alcohol concententation may be under .10% even though the breathalyzer gives a reading over .10%.

Experience has shown that repeated breathalyzer measurements taken on the same person produce a distribution of responses that can be described by a normal distribution with mean equal to the person's true blood alcohol concentration and standard deviation equal to .004%.

Problems:

i)  Suppose that a driver is stopped on his way home from a party. He has a true blood alcohol concentration of .095%, barely below the legal limit. If he takes the breathalyzer test, what are the chances that he will be incorrectly booked on a DUI charge (i.e., his result will be above .10%)?

ii) Suppose a different driver is stopped on his way home and is wicked drunk with a blood alcohol level of .15%. What are the chances that the breathalyzer will indicate that she is not  guilty of DUI?

iii)  In one bad night, 9 people from the same party are pulled over  by the police and given breathalyzer tests. Let's assume that all of them have a blood alcohol content of  .10%. What is the probability that at least one of the people will be booked on a DUI charge?

2.  Let's win some money!!

You can choose to play one of two games.  Each game costs one dollar to play.

Game 1:

A wheel with three numbers on it--zero, one, and two--is spun so that there is a 40% chance that the wheel lands on zero, a 10% chance the wheel lands on one, and a 50% chance the wheel lands on two.   You get back the amount in dollars of the number that the wheel lands on.

Game 2:

A different wheel with three numbers on it--zero, one, and two--is spun so that there is a 5% chance that the wheel lands on zero, a 80% chance the wheel lands on one, and a 15%chance the wheel lands on two.   You get back the amount in dollars of the number that the wheel lands on.

Problems:

(i)  What is the expected amount of money that you will get back from Game 1?    What is the expected amount of money that you will get back from Game 2?  (Don't forget to answer both of these questions.)

(ii)  If you played these games every second of every day for the rest of your life, which statement is most likely to be true:  a) you will make more money playing Game 1; (b)  you will make more money playing Game 2; or, (c)  you will make the same amount of money playing either game.   Justify your answer in two or less sentences.

(iii)  Which game has the larger standard deviation in returns?  Justify by calculating the standard deviations of the return amounts in both games.

(iv)  In Game 1, what is the probability that you will not lose money?   In Game 2, what is the probability that you will not lose money?

See the answers for a discussion of when each game is comparitively favorable.

3.  Stock Returns

Here is a problem taken from the 1988 Chartered Financial Analysts examination, which people working in finance need to pass to make lots of money.

Assume that stock in the Anheuser-Busch Companies, Inc.,  returns the following percentages:

Annual Return Rate         Probability
----------------------------------------------
    20%                                      .20
    50%                                      .30
    30%                                      .50

Let R = the annual return rate.

Problems:

1.  Calculate the expected value of the return rate.
2.  Calculate the standard deviation of the return rate.

Stocks with large standard deviations have return rates that can swing wildly from year to year, and so their owners can make or lose lots of money.   Stocks with smaller standard deviations have return rates that are typically closer to their expected values.   Many mutual fund companies build their stock portfolios by balancing stocks with large and small variances.

Source:
Bodie, Z., Kane, A., and Marcus, A.   Investments (2nd Edition).  Boston: Irwin, 1993, p. A-4.
 

4.  Hack-a-Shaq

Shaquille O'Neal is one of the best players on the professional basketball team the Los Angeles Lakers.   Shaq, as he is nicknamed, stands 7' 1" tall and weighs 330 pounds.  Most of the shots he takes are close to the basket, and because he is so big other players have a hard time stopping him from making baskets.  In fact, he makes 57.2% of  his shots, which is impressive given that most players make about 45%.

In basketball,  when a player trying to make a shot is hit on the body by someone on the opposing team, thereby causing the player to miss the shot, the player gets to take two free shots from 15 feet away from the basket.  These shots are called foul shots.   Shaq does not shoot foul shots very well.  In fact, he makes only 51.3% of his foul shots.

Regular shots are worth two points.  Foul shots are worth one point each.

Because Shaq is less likely to make foul shots, one strategy is to foul him whenever he touches the ball.  This strategy has been nicknamed the "hack-a-Shaq."  Let's see if the hack-a-Shaq pays off.

Problems:

1.  Calculate the expected value of the number of points Shaq scores on one regular shot (not foul shots) at the basket (i.e., he makes or misses the shot).   Regular shots are the ones he has a 57.2% chance of making.

2.  Assume that all foul shots are independent events (examinations of foul shooting records suggest this is approximately true).  Calculate the expected value of the number of points Shaq gets when he shoots two foul shots.

3.  Compare the EV for  foul shots to the EV for regular shots.   Based on these expected values, should opposing teams adopt the hack-a-Shaq?

Source: http://www.espn.go.com/nba/teamstats?team=lal

5.   Problem on the Central Limit Theorem

A bottling company uses a filling machine to fill plastic bottles with cola.  The bottles are supposed to contain 300 milliliters (ml).  In fact, based on millions of measurements of bottles, the company knows that the contents of any particular bottle vary according to a normal distribution with mean 298 and variance 9.

Problems:

i)  What is the probability that an individual bottle contains less than 295 ml?

ii) What is the probability that the average content of the bottles in a six-pack is less than 295 ml?

iii) What is the probability that less than 10% of the bottles in a case of 100 bottles have content less than 295 ml.

iv) What is the 99th percentile for the number of milliliters of an individual bottle (i.e., the number such that 99% of all bottles have less than this amount)?