Probability Project

Probability Project

Following this introduction are several probability problems. You are to choose any of 3 of these problems to work. They will be worth 30 points each for a total of 90 points you may do one extra for 10 bonus points. They will be due the day after the Chapter 6 test.

If you are asked to perform a simulation you must describe your simulation (remember the 4 steps!) and record each simulation. For example, if you were playing craps, you might state one game like this: 6-4, 2-1, 3-6, 4-2, 3-4 LOSS. I want to be able to see what you have done, rolled, drawn or whatever. You may chose to actually "play" some of the problems instead of simulating them. That is fine. Just include the outcomes of each "trial" as you would a simulation. You should still use the 4 steps. You may include these listings in an appendix. Simply coming up with the "correct" final answer is not sufficient for a perfect grade. Presentation, documentation, and well-explained conclusions are crucial. This does not need to be typed, but neatness is very important.

PROBLEMS:

Cereal Toys: A certain brand of cereal is running a promotion where you can get 1 toy in each box purchased. There are 5 different toys. I wish to collect one set of these toys to give as a gift. Assuming the toys are randomly distributed among the boxes, how many boxes should I expect to buy before I have all 5 toys? Perform a simulation and repeat it 20 times. What is the average number of boxes you needed to buy before a set was obtained?

Dice Rolling: Italian gamblers used to bet on the total number of dots rolled on three regular six-sided dice. They believed the chance of rolling a nine ought to equal the chance of rolling a total of 10 since there were an equal number of different ways to get each sum. However, experience showed that these did not occur equally often. The gamblers asked Galileo for help with the apparent contradiction, and he resolved the paradox. Can you do the same? Please answer the question in a well-written paragraph or series of paragraphs. Be sure to explain both why the gamblers were confused and what the actual probabilities are. You may include evidence from actual trials, simulations, and theoretical calculations.

Chuck-A-Luck: The game Chuck-A-Luck is played as follows: A player bets $M on an integer chosen from 1- 6 and then rolls three dice. If the number appears on exactly one die, then the player wins the amount bet. The player wins twice the amount bet if the number appears on two dice and three times the amount bet if the number appears on all three dice. If the number does not appear, then the player loses the $M. What is the probability of winning at each level? At any level? For every n such bets of a fixed amount $M, what would be the House's average net gain? (In other words, find  , such that  = average gain by the house.) Calculate the theoretical value. Verify this value repeating a simulation 50 times for M=$1.

Cut the Cards: Suppose we take a shuffled deck of 52 cards and make three cuts into the deck, where each successive cut is deeper than the previous. What is the probability p of getting an Ace, a Deuce, or a Jack within these three cuts? Derive the probability mathematically. Describe a simulation and perform it 30 times.

Dice Rolling #2: [In 17th century France, the Chevalier de Mere posed a famous problem which led to the axiomatic development of probability theory. Blaise Pascal first solved the problem in collaboration with Pierre de Fermat.] Determine which of the following two situations will give you better odds (if either). Determine the solution mathematically and then perform a simulation or trial. Do twenty sets for each "game." (1) If you roll a fair die over and over until a Six is rolled, what is the probability of rolling the Six within 4 rolls? (2) If one rolls two dice, what is the probability of rolling a Double Six within 24 rolls?

The Matching Problem: Suppose m addressed letters are to be stuffed into m pre-addressed envelopes. If the letters were stuffed at random over a period of many days, what would be the average number of correct matches? Describe a simulation. Repeat it five times for each of 4 different values of m where m ≥ 5.