Exercises and Projects for Chapter 4: Probability

1. Assume you flip a coin and get $3 if the result is heads, and have to pay $2 if the result is tails. What is the expected value?
2. You face 5 envelopes, containing $0, $1000, $2000, $3000, and $4000. You randomly choose one of them. What is the expected value?
3. You have five $1-bills, three $5-bills, and one $10 bills in your pocket. You randomly choose one of them to give the taxi driver a tip. What is the expected value?
4. You want to buy a used car tomorrow, for this reason you went to your bank and took $8000 in cash. In the evening there is a football game which you want to attend, you already have the ticket worth $15. You read in the newspaper that during the last game, 5 out of 80000 spectators were stolen their money. You are living in your apartment since 10 years and never had a burglary. What do you do? 1) Stay at home and watch your money, or 2) go to the game and leave the money at home, or 3) go to the game and take the money with you? Explain.
5. a) Two cards are selected randomly from a shuffled 52-card deck. Someone tells you that the first card is black. What is the probability that the second card is black too?
   b) Again two cards are selected randomly from a shuffled 52-card deck. Someone tells you that at least one of the two cards is black. What is the probability that both cards are black?

6. SENATE RACE(compare [Kockesen]) An incumbent senator decides first whether to run an expensive ad campaign for the next election. After that, the challenger decides whether to enter the race or not. The chances for the senator to win are 5/6 with the ad campaign, and 1/2 without. The value of winning the election is 2, of losing -0.5, and the cost of the add campaign is 1. (all values in million dollars). Analyze the game.
7. a) You put $100 on "red" in European roulette. If you lose, the bet is gone. If you win, you get your bet back, plus additional $100. How much total win or loss would you expect?
   b) You put $10 on the numbers 1, ... 12. If you lose, you lose the bet. If you win, you get your bet back plus additional $20. How much total win or loss would you expect?
   c) You put $5 on the number 13. If you lose, you lose the bet. If you win, you get your bet back plus additional $175. How much total win or loss would you expect?
8. You want to insure your yacht, worth $90,000. A total loss may occur with probability 0.005 in the next year, a 50% loss with probability 0.01, and a 25% loss with probability 0.05. What premiums would you have to pay if the insurance company wants to expect a profit of $200 per year?
9. A random number generator produces a sequence of three digits, where each one of the digits 0, 1, 2, and 3 has equal probability of 1/4. Each digit is generated independently of the others.
   Draw the probability tree for this three-step experiment.
   Find the probability that a sequence
   a) consists of all ones, or
   b) consists of all odd digits.
10. Three digits are generated in three rounds. In the first round, the leading digit is selected. It is either 4 or 5, with equal probability. In each further round, a digit is selected among the digits larger or equal than the previously selected one, with equal probability. That means, if the first digit selected was a 5, the next one could only be one of the digits 5, 6, 7, 8, 9, which all have probability 1/5. If let's say 7 is selected as second digit, then the third one must be one of 7, 8, 9, all having equal probability of 1/3, and so on.
    Draw the probability tree for this three-step experiment.
    Find the probability that a sequence
    a) consists of all even digits, or
    b) consists of all odd digits.
11. There are eight balls in an urn, identical except the color. Three of them are blue, three of them red, and two of them green. Consider the three-step experiment of
    • drawing one ball, setting it aside,
    • drawing another ball, and setting it aside, and
    • drawing a third ball, and setting it aside.
    a) Draw the Probability Tree or Probability Digraph for this three-step experiment.
    b) How likely is it that two of the drawn balls are blue, and one of them is green?
    c) How likely is it that all three balls have different colors.
    d) How likely is it that all three balls have the same color.
12. a) Draw a probability tree for all the possible head-tail sequences that can occur when you flip a coin four times.
    b) How many sequences contain exactly two heads?
    c) How many sequences contain exactly three heads?
    d) Draw the probability digraph for the case where you are only interested in how many heads a sequence occurs, not when they occur. That is, you would identify the situations HT and TH, you would identify HHT and HTH and THH, and so on.
    e) What is the probability to get exactly two heads in a sequence of four attempts?
13. There are eight balls in an urn, identical except the color. Four of them are blue, two of them red, and two of them green. Consider the three-step experiment of
    • drawing one ball, setting it aside,
    • drawing another ball, and setting it aside, and
    • drawing a third ball, and setting it aside.
    a) Draw the Probability Tree or Probability Digraph for this three-step experiment.
    b) How likely is it that two of the drawn balls are red, and one of them is green?
    c) How likely is it that all three balls have different colors.
    d) How likely is it that all three balls have the same color.
14. There are eight balls in an urn, identical except the color. Four of them are blue, three of them red, and two of them green. Consider the three-step experiment of
    • drawing one ball, setting it aside,
    • drawing another ball, and setting it aside, and
    • drawing a third ball, and setting it aside.
    a) Draw the Probability Tree or Probability Digraph for this three-step experiment.
    b) How likely is it that two of the drawn balls are blue, and one of them is red?
    c) How likely is it that all three balls have different colors.
    d) How likely is it that all three balls have the same color.
15. A coin is flipped at most 5 times. When a "Tail" shows, the experiment is stopped, otherwise, and if the coin hasn't been flipped more than 4 times, the coin is flipped again.
    a) Draw the probability tree of the multistep experiment.
    b) What is the probability that exactly one "Tail" occurred?
16. You draw 3 cards out of a shuffled 32 cards deck. What is the probability to have a straight--- a sequence of three consecutive cards, like 10, J, Q?
17. ** You draw 3 cards out of a shuffled 32 cards deck. What is the probability for a triple---all of them having the same rank? What is the probability to have a pair but not a triple?
18. You draw 3 cards out of a shuffled 32 cards deck. What is the probability for a flush---all three cards having the same suit?
19. ...

1. TENNIS: Steffie and Arantxa are playing a tennis set. Recall that a set consists of games and games consist of points. A game is won by the player who has first won at least four points in total and at least two points more than the other player. A set is won by the player who has won at least six games, and at least two more than the other player. Assume each single point is won by Steffie with probability 0.52 and by Arantxa with probability 0.48. How likely is it that Steffie wins a game? How likely is it that Steffie wins the set? Does it matter whether we play with or without tie-break?
2. Project: FINAL EXAM: This game is played by three players: The teacher, student X and student Y. Of course, there are also 20 other students, but they don't make decisions and therefore do not really participate in the game. Student X makes the first move by wearing his cap to class or not. Seeing this, Student Y has the same option, cap or not. Seeing the two students with or without caps, the teacher makes the decision to have a special eye (a) on X, or (b) on Y, or (c) the teacher could also relax. Seeing the teacher react, each one of students X and Y decides for himself whether to cheat or not.
   Cheating, when not noticed by the teacher, brings a better grade, a payoff of 1. If the teacher notices cheating, disciplinary measures are taken, a payoff of -6 for the student. Cheating is noticed by the teacher

   • with probability 1/5 if the student has no cap to hide and the teacher has no special eye on the student,
   • with probability 1/10 if the student wears a cap but the teacher has no special eye on the student,
   • with probability 1/3 if the student wears no cap and the teacher has a special eye on the student,
   • with probability 1/6 if the student wears a cap and the teacher has a special eye on the student.

   The teacher has also payoffs. It is the sum of a value of -1 if he or she has to keep an eye on a student, compared to 0 if he or she can relax, and a value of -2 if cheating has occurred but gone unnoticed.
   How will the players play? Modify the model if you deem it necessary.
3. .....