Homework 3 - Probability (Due October 1st at the beginning of class)

Your assignment should be prepared in Word or a comparable program. Except in unusual circumstances, submit your assignment in one of the following two ways:  1) handing it to me in person in class or elsewhere or 2) putting in my mail box in SAC 268 (go into the office and turn right where you will find some mailboxes just inside the door). In exceptional situations I will accept email submissions, but please make arrangements with me in advance.

1. You are at a casino with a friend, playing a game with which dice are involved.  You friend has just lost six times in a row.  He is convinced that he will win on the next bet because he claims, by the law of averages, it's his turn to win. He explains to you that the probability of winning this game is 40%, and because he has lost six times, he will probably win the next four.  Is he right? Explain briefly.

2. Suppose a friend reports that he has just had a string of "bad luck" with her car.  She had three major problems in as many months and now has replaced many of the worn parts with new ones.  She concludes that it is her turn to be lucky and she shouldn't have any more problems for a while.  Is she using the Gambler's Fallacy?  Briefly explain.

3. A few years ago, Taco Bell had a scratch-off card promotion that contained four separate games.  The player could play only one of the them and could choose which one.  The probability of winning game A was 1/2, and the prize was a drink worth 55 cents.  The probability of wining game B was 1/4, and the prize was a food item worth 69 cents.  The probability of winning game C was 1/8, and the food prize was worth $1.44. The probability of winning game D was 1/16, and the food prize was worth $1.99.

a.  Calculate the expected value of each game.  (Note you didn't have to pay for the card, so the payoff if you don't win is simply 0.) Which game had the highest expected value.

b. For what reasons would people choose to play the other games even if they knew the expected values were lower?

4.   In Illinois's "Little Lotto" lottery game, you choose five numbers between 1 and 39.  Five numbers are then randomly chosen and you receive a prize as given in the following table.  A Little Lotto lottery ticket costs $1.

* It turns out that if the first prize is not awarded on a given drawing, a portion of it is carried over to the next drawing.  We are going to ignore this detail of the game. It actually makes little difference in the calculation!

a. What is the probability of not winning anything? (I.e. what is the value of the question mark in the table?)

b. Calculate the expected value of a single play of Little Lotto?  (Hint: you need to use payout column.)

c.  Simulate playing Little Lotto 30000 times in Excel. (To do this, make a table like

over to the side.  Then as usual simulate 30000 trials.)  How many times did you lose your $1?  How many times was the payoff  $0, $9, $99, and $99,999?

d. Little Lotto drawings occur 365 days a year.  About 350,000 tickets are sold per day.  Using the expected value you calculated in part b, determine the approximate income derived for the State of Illinois each year from Little Lotto.

e. According to the Illinois State Lottery, approximately 77% of the gross income from the lottery (the money not returned as prizes) goes to support public schools; about 16% goes to vendors who sell the tickets; and about 7% pays for operating expenses and advertising.  Using the number you calculated in part d), calculate approximately how much of the income  the Little Lotto generates each year goes in each category.     (By the way, there are many other official state lottery games in Illinois, and the Little Lotto doesn't even generate one-tenth of the total income generated by the state lottery. It is a big business.)

5. Even highly accurate medical tests can be lead to surprisingly high false positive rates.  Consider a standard test for Lupus, which is a chronic (long-lasting) autoimmune disease in which the immune system, for unknown reasons, becomes hyperactive and attacks normal tissue.  Lupus is thankfully rare: it occurs in 33 people per 100,000.  A standard test for Lupus is 94% accurate in detecting Lupus and is 97% correct in identifying people who don't have it.  Using the same technique as in Activity 4, calculate the false positive rate for this test.