Project 11.1

What do the following people have in common?
Paul Painleve, President of France
Omar Khayyam, author of The Rubaiyat
Emanual Lasker, world chess champion
James Moriarty, Sherlock Holmes's nemesis, author of The Dynamics of an Asteroid

Project 11.2

Devise a fair scheme for eliminating coins in this country.

Here is an alternative question:
Michael Rossides has come up with a scheme for eliminating coins. This scheme involves probability and the fact that most cash registers today are computers. Suppose that every cash register could be programmed with a random number generator; that is, suppose that it were possible to pick a random number from 1 to 99. Rossides' system works as follows. Suppose you purchase items totaling $15.89. The computer would choose a number from 1 to 99 and then compare it with the cents portion of the purchase. In this case, if the random number is between 1 and 89 the price would be rounded up to $16; if it is between 90 and 99 the price would be rounded down to $15. For example, if you purchase a cup of coffee for $1.20, and the random number generator produces a random number from 1 to 20, the price is $2, but if it produces a number from 21 to 99 the price is $1. Write a paper commenting on this scheme.

Project 11.3

Find the probability that the 13th day of a randomly chosen month will be a Friday. (Hint: It is not correct to say that the probability is 1/7 because there are seven possible days in the week. In fact, it turns out that the 13th day of a month is more likely a Friday than any other day of the week.) Write a paper discussing this problem.

References:

William Bailey, "Friday-the-Thirteenth," The Mathematics Teacher, Vol. LXII, No. 5 (1969), pp. 363-364.
C. V. Heuer, Solution to Problem E1541 in American Mathematical Monthly, Vol. 70, No. 7 (1963), p. 759.
G. L. Ritter, et al, "An Aid to the Superstitious," The Mathematics Teacher, Vol. 70, No. 5 (1977), pp. 456-457.

Project 11.4

The questions in this problem are from a study by MacCrimmon, Stanbury, and Wehrung, "Real Money Lotteries: A Study of Ideal Risk, Context Effects, and Simple Processes," in Cognitive Processes in Choice and Decision Behavior, edited by Thomas Wallsten, (Hillsdale, N. J.: Lawrence Erlbaum Associates, 1980, pp. 155-179).

Question: You have five alternatives from which to choose. List your preferences for the alternatives from best to worst.

1. sure win of $5 and no chance of loss
2. 6.92% chance to win $20 and 93.08% chance to win $3.98
3. 27.52% chance to win $20 and 72.48% chance to lose 69 cents
4. 61.85% chance to win $20 and 38.15% chance to lose $19.31
5. 90.46% chance to win $20 and 9.54% chance to lose $137.20
a. Answer the question based on your own feelings.
b. Answer the question using mathematical expectation as a basis for selecting your answer.
c. Conduct a survey of at least 10 people and summarize your results.
d. What are the conclusions of the study.

a. Answer each of the following questions* based on your own feelings.
b. Answer questions (1) and (2) using mathematical expectations as a basis for selecting your answers.
c. Conduct a survey of at least 10 people and summarize your results.
1. Choose between A and B:
A. A sure gain of $240
B. A 25% change to gain $1,000 and a 75% chance to gain $0
2. Choose between C and D:
C. A sure loss of $700
D. 75% chance to loose $1,000 and %25 change to lose nothing
3. Choose between E and F
E. Imagine that you have decided to see a concert and have paid the admission price of $10. As you enter the concert hall, you discover that you have lost the ticket. Do you pay $10 for another ticket?
F. Imagine that you have decided to see a concert where the admission is $10. As you going to enter the concert ticket line, you discover that you have lost one of your $10 bills. Would you still pay $10 for a ticket to the concert?
* These questions are from A Bird in the Hand, by Carolyn Richbart and Lynn Richbart in The Mathematics Teacher, November 1996, pp. 674-676.

Project 11.6

Do some research on Keno probabilities. Write a paper on playing Keno.

Reference:

Karl J. Smith, "Keno Expectation," Two-Year College Mathematics Journal, Vol. 3, No. 2, Fall 1972.