This assignment is due at the beginning of your discussion section on Wednesday, November 19.
Summary: This assignment will give you practice with some basic probability concepts.
Feel free to discuss these problems with others, but be sure you know how to solve them (and similar problems) on your own.
(a) Every day you download some investment information from a web site whose server's load varies on different days of the week. On Monday through Thursday, the download takes you 15 minutes. On Friday, it takes you 25 minutes. On Saturday and Sunday, it takes you 10 minutes. What is your overall expected download time (i.e., imagine you don't know what day it is)?
(b) A standard slot machine has three wheels, each with 20 symbols distributed as shown below. When the arm is pulled, each of the 20 symbols on each wheel is equally likely to come up.
|Symbol||Wheel 1||Wheel 2||
(b.2) What is P(a bar comes up on Wheel 2)?
(b.3) What combination of symbols on the three wheels should yield the highest payoff (assuming payoffs decrease with higher probabilities)? There are two equivalent answers to this.
(b.4) What is P(a lemon comes up on all three wheels)?
(b.5) What is P(not getting a plum on Wheel 1)? (Complementary probabilities may help here.)
(c) In the California Lottery game SuperLotto, a player would pay $1.00 to pick six numbers (between 1 and 51) for the next draw. (We're describing the original SuperLotto game, not the current SuperLotto Plus with a separate "Mega" number; the probabilities for SuperLotto Plus are a little more complicated.) Every Wednesday and Saturday, the Lottery draws six numbers. If the player's six numbers match the Lottery's six numbers, the player wins the multi-million-dollar jackpot (or splits it with any other players who also picked the same six winning numbers). If nobody matches the six winning numbers, the jackpot "rolls over" to the next draw; that is, the jackpot amount for the draw with no winners is added to the jackpot for the next draw. If many draws go by with no winners, the jackpot can get very large; it has been over $100,000,000.
Which of the following statements are supported by the principles of probability? Give a yes or no answer to each, with a few words of explanation.
(c.1) If you pay $2.00 for two different sets of numbers, you are twice as likely to win as if you paid $1.00.
(c.2) If you pay $100.00 for 100 different sets of numbers, you are 100 times as likely to win as if you paid $1.00.
(c.3) If you buy one ticket for every drawing for ten years, your chances of winning are roughly a thousand times greater than if you buy just one ticket.
(c.4) If you decide to play the same six numbers in every draw from now on, you should check the winning numbers in the past to be sure your numbers haven't come up already.
(c.5) If you play for a few months and not a single number you choose is included in the winning numbers, you are a little more likely to win the next draw (because you're "due")
(c.6) If you play for a few months and two or three of your numbers are included in the winning numbers of each draw, you are a little more likely to win the next draw (because you're "on a roll").
(c.7) Since 50% of the ticket revenues goes to prizes, in general the expected value of a $1.00 ticket is 50 cents.
(c.8) The expected value of your $1.00 ticket will will be higher if you only play when the jackpot is over $10,000,000.
(c.9) Your probability of winning is greater if you pick numbers between 1 and 31 (because many people pick birthdates as their numbers).
(c.10) Your expected value is greater if you pick numbers greater than 31.
(c.11) (extra credit--we won't ask you to do something like this on an exam) Calculate the probability of winning the SuperLotto jackpot.
(c.12) (extra credit) Calculate the expected value of playing a $1.00 SuperLotto ticket. The hard part about this isn't the calculation; it's finding the payout amounts over the last year. They're not available at the California Lottery web site (www.calottery.com), and we don't know if they're available anywhere, so this is an opportunity for you to do some creative web searching.
(d) [from Patterns of Problem Solving by Moshe F. Rubinstein] In one brief sentence, how is information related to probability?
If you know that 5 people out of every 1000 have cancer, and if we have a perfectly accurate test that predicts whether a person has cancer, which of the following gives us more information:
* The test indicates that a person has cancer.
* The test indicates that a person does not have cancer.
What to turn in: A word processing document that includes the answers to all the questions in each part above. Grading depends on completeness, thoroughness, correctness, and clarity.
Written by David G. Kay, Summer 1999; revised Fall 1999, Fall
2000, Fall 2001, and Fall 2003.
Problem (b) was adapted from Harold Jacobs' text, Mathematics: A Human Endeavor.
Problem (d) comes from Moshe Rubinstein's text, Patterns of Problem Solving.
David G. Kay, 406B Computer Science
Tuesday, November 18, 2003 -- 4:41 PM