CMPSCI 240: Reasoning About Uncertainty

Second Midterm Exam

David Mix Barrington

13 March 2009

Directions:

• Answer the problems on the exam pages.
• There are six problems for 100 total points. Actual scale is A=88, C=60.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first four questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• The parts of Question 5 have numerical answers -- you may give your answer in the form of an expression using arithmetic operations, powers, falling powers, or the factorial function. If you give your answer using the "choose" notation, also give it using only operations on this list. In addition, if your answer is a non-negative integer less than or equal to 100, you must compute the number for full credit.
• The paper exam sheet also contained the first twelve rows of Pascal's triangle.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 20 points
  Q6: 40 points
  Total: 100 points

• The first two true-false questions involve a situation where we repeatedly roll a single fair six-sided die until all six numbers have come up at least once.

• Question 1 (10): True or false with justification: The expected number of times we throw the die, before all numbers have come up at least once, is more than twelve.

• Question 2 (10): True or false with justification: The probability that the six numbers will all occur within the first six throws is less than 5%.

• The next two true-false questions involve a game of Texas Hold'em, where a player has a two-card hand, forms a seven-card hand with the five cards on the table, and chooses five of those seven cards to be her poker hand. We ignore the cards held by any other players. (Thus we assume that the player's two cards are chosen uniformly from the 47 cards not on the table.)

• Question 3 (10): True or false with justification: If the five cards on the table have five different ranks, the probability that our player's seven cards include three of a kind is 5*(3 choose 2)/(47 choose 2).

• Question 4 (10): True or false with justification: If the five cards on the table have five different ranks, the probability that the player's seven cards have seven different ranks is strictly less than (32 choose 2)/(47 choose 2).

• Question 5 (20): In an illegal lottery run by organized criminals, a player may bet $1 on a three-digit number (possibly with leading zeros) and wins $600 if this is the number chosen. All three-digit numbers are equally likely.
  • (a,5) What are the expected winnings from this $1 bet? (Do not count the $1 paid to bet.)
  • (b,5) What is the probability that the three-digit number has three different digits?
  • (c,10) Suppose that I am secretly told, before I bet, whether the three digits are all different. I then choose a number to bet either from the all-different-digit numbers, or from the not-all-different-digit numbers, with all such numbers being equally likely. What are the expected winnings from my bet on a random all-different-digit number, given that the actual number has all different digits and that the prize is still $600? What are the expected winnings from my bet on a random not-all-different-digits number, given that the actual number is of this tyoe and that the prize is $600?

• Question 6 (40): This problem again involves Zane's Noodle Bowl, which offers 13 kinds of vegetables that may be put in your soup.
  • (a,5) Customer A likes to choose her vegetables randomly, so that each set of vegetable types is equally likely. How many such sets are there (including sets of all sizes from 0 through 13)? How may she easily choose a set randomly, using coins, cards, or dice?
  • (b,5) Customer B always wants exactly five of the 13 vegetable types, but would like each set of five vegetables to be equally likely. How many five-element sets are there? How can he easily choose such a set randomly, using coins, cards, or dice?
  • (c,10) Let NV be the random variable giving the number of vegetables chosen by Customer A using the procedure in part (a). What is the probability that NV is equal to either 6 or 7, that is, Pr((NV = 6) ∨ (NV = 7))? Give an expression for the exact answer, and a numerical approximation accurate to plus or minus 0.05. You may or may not find the Pascal's Triangle on the paper test sheet to be helpful.
  • (d,10) Compute the mean and variance of the random variable NV. (Hint: It may be useful to break NV up as the sum of other random variables.)
  • (e,10) Using the Normal Approximation to the Binomial, find a number k so that the probability is at least 95% that the average value of NV, for k separate visits to Zane's by Customer A, is between 6 and 7. Justify your answer.

Last modified 14 March 2009