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## Problems: Drawing Tickets from a Box Part 2

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13. An insurance company selling term life insurance finds that one in 99 people who buy the policy will die. The company must then pay out \$20,000. It sells the insurance policies for \$300. If we put this situation into the box model that the book develops,

a) How many tickets should be in the box?
b) How many of those tickets have a \$300 on them?
c) You might think that there is at least one ticket with a -\$20000 on it, but there is not. What is the correct amount and how many tickets have that amount?

14. Suppose you are gambling on a coin flip. When a head comes up, you win a dollar and when a tail comes up, you lose a dollar. You play this game 100 times.

a) Playing this game 100 times is like taking 100 draws from the box _______
b) If you play the game 100 times, the most you could lose is ________ and the most you could gain is _________.
You would expect to break even if you played this game 100 times, but by random chance you might end up winning some or losing some. It turns out that we can estimate quite accurately your chances of winning various amounts using a normal curve with an average of zero and a standard deviation of 10. (This week we will see why!)
c) If you play the game 100 times, approximately what is the probability that you will lose \$15 or more?
d) (A working backwards problem) You have a one percent chance of winning ________ or more.

15. A box has five tickets, 1, 2, 3, 4, 5.

a. What is the average of the box?
b) What is the standard deviation of the box?
c) If 200 draws are made from the box, the sum should be __________ give or take ____________. (Find the expected value and standard error of the sum.)
d) If 200 draws are made from the box, the average of the draws should be __________ give or take ____________. (Find the expected value and standard error of the average.)
e) If 800 draws are made from the box, the sum should be __________ give or take ____________. (Find the expected value and standard error of the sum.)
f) If 800 draws are made from the box, the average of the draws should be __________ give or take ____________. (Find the expected value and standard error of the average.)

16. Joe and Ted are playing a game with two coins. If both coins come up heads, Joe pays Ted \$1.00. If both coins come up tails, Ted pays Joe \$1.00. If one coin is heads and one is tails, no payment is made. They play this game 98 times.

For Ted, playing this game is like taking _______ draws from the box __________. (Be careful.)
Ted's net gain from playing this game 98 times will be around _________ give or take _________ or so.

17. You are a casino offering a game with ten cards, the ace through the ten of hearts. The cards are shuffled and then a player randomly picks a card. If it is the five or hearts, you (the casino) pay out \$80. If it is any other card, you collect ten dollars. Explaining clearly how you get your answer, what is the probability that you will be losing money after 100 plays? After 400 plays? How many times would you have to play this game so that you have less than a 1% chance of losing money?

18. In question 1 we looked at the box containing these five tickets: 0,0,0,0,5. We found the mean was 1 and the standard deviation was 2. If we draw from this box 400 times, we will expect to get a total of 400 give or take about 40. If we could repeat this experiment a great many times, we would end up with a normal distribution with a mean of 400 and a standard deviation of 40.

If we draw 400 times from this box, we could get a total as low as zero and a total as high as 2000. What is the probability that our draw of 400 tickets from this box will total more than 500?

19. Chance Pickens pulls a card from a deck of cards. If it is an A, J, Q, or K, he writes a one on a piece of paper. If it any of the other nine cards, he writes a zero. He then shuffles the deck. If he repeats this process 100 times, the expected percentage of ones in his 100 entries is ________ percent give or take ________ percent. (You will need the short-cut formula for finding the standard deviation of a zero-one box. Let p = fraction of ones, and q = fraction of zeros. The standard deviation is the square root of p*q.)
With the mean and standard error you have just found, use the normal curve to approximate the probability that Chance will get more than 35% successes (writing down a one rather than a zero) next time he performs his strange exercise?

20. A dice game has the following rules. A single die is rolled and if a 1, 2 or 3 is rolled, you lose a dollar. If a 4, 5 or 6 is rolled, you win a dollar. If you play this game 150 times, you would expect to win \$__________ give or take \$___________.

a) What is the box?
b) What is the average of the box?
c) What is the standard deviation of the box?
d) What is the expected sum?
e) What is the standard error of the sum?
f. Your friend Joe claims that after he played this game 150 times, he came out ahead by \$39 dollars. What is the probability of getting \$39 or more from this game?

21. A dice game has the following rules. A single die is rolled and if a 1 is rolled, you lose a dollar. If a 2, 3, 4 or 5 is rolled, you neither lose nor win money. If a 6 is rolled, you win a dollar. If you play this game 150 times, you would expect to win \$__________ give or take \$___________.

a) What is the box?
b) What is the average of the box?
c) What is the standard deviation of the box?
d) What is the expected sum?
e) What is the standard error of the sum?
f. Your friend Joe claims that after he played this game 150 times, he came out ahead by \$39 dollars. What is the probability of getting \$39 or more from this game?

22. A dice game has the following rules. A single die is rolled and if a 1 or 2 is rolled, you lose a dollar. If a 3 or 4 is rolled, you neither lose nor win money. If a 5 or 6 is rolled, you win a dollar. `

a) How would you represent this game in terms of a box of tickets?
b) What is the average of the box?
c) What is the standard deviation of the box?
d) What is the expect mean?
d) If you play the game 150 times, what is the standard error of the sum?
e) If you play this game 150 times, you would expect to win \$__________ give or take \$___________. (Find the expected sum for the first blank and the standard error of the sum for the second.)
f) Your friend Joe claims that after he played this game 150 times, he came out ahead by \$39 dollars. Do you consider his claim believable or not? Explain.

23. A person spins a wheel that has an equal chance of landing on any one of five colors. If it lands on white the person wins \$1, on red \$3, on blue \$4, on green \$5, and on yellow \$7. (Note, this is like picking a ticket from a box with these five tickets: 1, 3, 4, 5, 7.)

a) What is the average of the box?
b) What is the standard deviation of the box?
c) If the person spins the wheel 100 times, he would expect to win \$____________ give or take about \$_____________. (In the first blank you need the expected value of the sum of 100 draws from the box, and in the second the standard deviation of the distribution we get from drawing 100 times from the box.)
d) What is the probability that this player will win more than \$450 in those 100 plays?

24. A box with a million tickets has an average of 76 with a standard deviation of 10. If a sample of 400 draws is made from this box, we expect an average of ______________ give or take ____________or so. The probability that the average would be more than 77 is ________________________.

25. A box has an average of 12 with a standard deviation of 3. One hundred draws are made from the box with replacement and the total is 1173.

This total represents the total of an expected sum of _______ plus a chance error of _____________. The z-score of the total is ____________.

26. Suppose that there is a risky investment that has a 90% chance of giving you a 20% return but 10% of the time it will lose all the money you invested. Imagine investing \$100 in this investment.

a) If we convert this to the box model, what tickets would be in the box?
b) What is the average of the box?
c) What is the standard deviation of the box?
d) Suppose that we put \$100 into two of these investments. The possible outcomes are lose 200, lose 80, and gain 40. What are the probabilities for each of these outcomes? What is the average outcome?
e) Suppose we have 100 of these investments that are independent of one another. (This means that if one is a loss, it does not affect the probability that the others will be losses.) If we invest \$100 in each of these investments, we would expect to gain ______ give or take about _______.
f) What is the probability that you would lose money with these 100 investments?
g) Suppose we had 400 of these investments. If we invest \$100 in each of these investments, we would expect to gain ______ give or take about ________.
h) What is the probability that you would lose money with these 400 investments?
i) In the real world it is dangerous to make the assumption that investments are completely independent of one another. Why?

27. Suppose that the show Deal or No Deal has a contestant who has the following three amounts still on the board: \$75, \$50,000, and \$400,000. If the offer from the banker is \$130,000, is the offer above or below the fair offer? By how much?

28. In the game Deal or No Deal, the contestant begins by choosing one case from 26 cases with the following amounts:

\$.01, 1, 5, 10, 25, 50, 75, 100, 200, 300, 400, 500, 750, 1000, 5000, 10000, 25000, 50000, 75000, 100000, 200000, 300000, 400000, 500000, 750000, 1000000.

a) Suppose that you want to illustrate this distribution with a histogram with five classes. What would it look like? (I am not sure how you can do this on-line, other than describing the classes and telling how many are in each class.)
b) What is the median amount of this distribution?
c) What is the fair value (the mean) of this game?
d) Suppose that the contestant opens these six boxes: \$1, \$50, \$100, \$750, \$50,000, and \$200000. Suppose that the banker makes an offer of \$46000. How much is this below the fair offer?

29. Suppose a slot machine that cost \$1.00 to play will on average pays out \$95. This means that the expected value or average of each play is -\$.05. Suppose further that the standard deviation of each play is \$1.50.

a) If a person plays this machine 100 times, she would expect to lose about ___ give or take ___.
b) If a person plays this machine 400 times, we would expect to lose about ____ give or take ____.
c) What is the probability that a person would come out winning money if she played the machine 400 times?

30. A spinning wheel has five colors, each equally likely to come up. If the wheel stops on red, the player loses \$3.00. If it stops on yellow, the player loses \$1.00. If it stops on black, the player neither loses nor gains. If it stops on green, the player wins \$1.00, and it if stops on blue, the player wins \$3.00.

a) If we put this situation into the box model, what is the box?
b) What are the average of the box and the standard deviation of the box?
c) If the player plays 144 times, the expected average amount he will win is \$____ give or take \$_____.
d) Using the normal curve, approximately what is the probability that a player playing this game 144 times will average winnings more than 33 cents?

31. A box has these eight tickets: 1, 1, 1, 3, 3, 3, 5, 7. The average of the tickets is 3. What is the standard deviation of this set of tickets?

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32. Today in class we played a game based on Deal-No Deal. We had ten
envelopes with these amounts:

1
2
3
4
5
10
25
50
100
200

Bart selected an envelope. Then he opened three of the remaining envelopes, which were 1, 5, and 25. The question was then what was the fair value of the game, or the fair offer to stop the game. Ten was the median, or middle value, but this understates the value because the high values are so large relative to the small ones. To get the fair value, we found the average or mean of the remaining values, or 369 divided by 7 or about 53. The banker, however, always begins with an offer much less than the fair value, so we offered 20.

The player then opened two more envelopes, 3 and 100. The new fair value was then 266 divided by 5, or still about 53. Since the banker offer usually gets closer to fair value, we increased the offer to 25.

Eventually the player ended with just two possibilities, 2 and 200. The fair value of this game is 202 divided by 2 or 101. The player rejected the offer of 95 and was left with 2.

In addition to the mean and median, we talked about graphing the possibilities to get a histogram, and then talked about how to measure the variation or dispersion of the numbers. The range is a quick and dirty way to talk about dispersion, but for mathematical reasons the standard deviation, which is not an intuitive measure, is used.

Suppose we have these three groups of numbers, all with the same average of 40:

39, 41
0, 80
0, 40, 80.

Which has the most dispersion and which the least?

To compute the standard deviation, subtract the mean from the numbers, then square the results, then add up these squared numbers, divide them by the number of numbers we have, and take the square root.

For the first, the answer is 1, for the second it is 40, and for the third it is 32.66. Is that what you get?

What do you get for these numbers: 0, 40, 40, 80?

33. Ninety-eight self-professed beer lovers were asked to identify a particular brand of beer in a blind taste test. There were 3 samples to choose from and 45 of them correctly identified the beer. Do these results differ from what we would expect to get by random chance? Explain.

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