Homework 8 (for lectures on 10/14,10/16)

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1 Fall 2014 MTH122 Survey of Calculus and its Applications II Homework 8 (for lectures on 10/14,10/16) Yin Su Topics in this homework: Topic 1 Discrete random variables 1. Definition of random experiments and random variables. 2. Definition of probability and its relation with frequency. 3. Know how to find the probability for simple random experiments. Topic 2. Expected value and variance 1. Definition and expected value and variance and their application. 2. The formulas of expected value and variance. Problems: 1. Which of the following events are random events: (1) when you review an unpublished book, the number of typographical errors in this book; (2) The number of students who are left-handed in a class of capacity 50; (3) The time needed to drive from Buffalo to New York City; (4) In a box of 5 blue card labelled as 1,2,3,4,5. Draw a random card from this box. The number on the card; (5) In a box of 5 blue card labelled as 1,2,3,4,5. Draw a random card from this box. The color of this card; (6) The number of jet accidents in the next year. (7) The area of a circle of radius 2 is 4π. (8) Pick a point (x, y) on the unit circle centered at the origin. The x coordinate; (9) Pick a point (x, y) on the unit circle centered at the origin. The distance from this point to the origin. 2. Suppose X, Y are discrete random variables with the following distributions: X Y Compute the expected value and variance for each random variable.

2 3. Consider the experiment of selecting a card from an ordinary deck of 52 playing cards (i.e. four suits, and face value A,2,3,,10,J,Q,K for each suit). Try to do the following: (1) The probability that a spade( ) is drawn. (2) The probability that an Ace is drawn. (3) The probability that a face card (J,Q,K s) is drawn. (4) Consider the suit of the drawn card as the random variable X. Finish the following probability distribution table: X spades ( ) clubs ( ) diamonds ( ) hearts ( ) (5) Divide the cards into three categories based on their face value: Aces, Number cards (face value 2-10), face cards (J,Q,K). Finish the table for probability of occurrence of each category. X Aces (A) Number cards (2, 3,, 10) Face cards (J,Q,K) (6) A gambling game states that: (a) If the card you draw is an Ace, you win $ 10; (b) If the card you draw is a face card (J,Q,K), you win $ 1; (c) If the card you draw is a number card (2-10), you lose $ 1 (or in other words, you win $( 1)). Now let Y be your potential profit. Finish the probability table: Y (7) Based on the game described in part (6), calculate the expected value of your profit. (8) Is the game in part (6) in the favor of the player? 4. (Side bet in three card poker game) A popular poker table game in casino is called the three card poker. In each game, there are two mode of play, namely Ante and Play, and Pair Plus. You can check the general rules online: We only consider the side bet Pair Plus here.

In the game, three cards in a 52-card deck are dealt to the player and the dealer. The player will win if the pattern he gets is better than or equal to a pair.

Suppose we also know the probability of the occurrence of each outcome (You don t need to know how this is computed. Actually higher probability theory is needed in the computation.

3 In the game, three cards in a 52-card deck are dealt to the player and the dealer. The player will win if the pattern he gets is better than or equal to a pair. (This side bet has nothing to do with the dealer s hand cards.) The winning patterns and payoff table are shown as follows. Suppose we also know the probability of the occurrence of each outcome (You don t need to know how this is computed. Actually higher probability theory is needed in the computation. We will just assume this is the correct probability. This result is also shown on the above Wikipedia page.) Hand card Pattern Payoff of occurrence Straight flush Three of a kind Straight Flush A Pair Others Now evaluate the average profit if you play this side bet once. Is the game against you? 5. (Game of Roulette Wheel, simplified) In the game of Roulette wheel, there is a layout with 37 single numbers (1 through 36, and 0) which correspond to a Roulette Wheel having identical numbers as the layout. The Dealer spins the Roulette Wheel in one direction and a small ball in the opposite direction. The ball will stop at one number (You can check the rules in Seneca Casino: com/sites/senecaniagaracasino.com/files/how-to-play/ roulette.pdf) Suppose now there are two options to play: (a) Mode of Straight up: place you bet on exactly one number out of the 37 numbers on the number when the wheel stops. This mode pays 35 to 1. (You will win $ 35 or you will lose $ 1 in one game). (b) Mode of odd or even: Select between odd or even numbers on the number when the wheel stops.

4 In this mode, 1, 3, 5,, 35 are odd, and 2, 4,, 36 are even. But 0 is treated as neither odd or even. This mode pays 1 to 1. (You will win $ 1 or you will lose $ 1.) Now let s evaluate the profit of these two games. Finish the following questions: (1) If we play straight up mode, what is the probability that a single number from 0 to 36 occurs in a game? (2) If we play straight up mode, the probability table of your profit is Profit in Straight up 35 (Win) 1 (Lose) (3) What s the expected return if we play straight up once? (4) If we play odd and even, compute the probabilities that odd occurs, even occurs or neither (0 occurs): Number occur Odd Even 0 (5) Find the probability table for your potential profit: Profit in odd and even 1 (Win) 1(Lose) (6) Compute the expected profit if we play odd and even once. (7) Comparing these two modes, which one has higher average profit (or lower average lost)? Does any of there two mode favor the player? 6. (Phone battery life) Phone company EPPLE considers two phone battery suppliers X and Y. EPPLE finds out the probability distribution of the battery life for the battery that each supplier produces: X (in hours) Y (in hours) Assume EPPLE company decides to pick the supplier whose battery has a longer life on average. Which supplier should it pick?

5 7. (Asset investment) You have two options to invest $ in the stock market. There are two nice stocks available, A and B. Suppose the current price of A and B are both $ 100. You are planning to buy 100 shares of all A, or 100 shares of all B. The possible payoff tables are shown: Price of A in a month Price of B in a month (1) Compute your probability distribution of your TOTAL profit from A or B: Profit of A in a month Profit B in a month (2) Compute the expected profit in a month respectively. Which stock has higher average profit? (3) Compute the variance of the profit from A and B respectively. Which stock has higher variance? (4) If you are assumed to be risk-averse, which stock will you buy? 8. (Road service package) BBB is a vehicle road service company. It provides emergency road service (such as towing, jump start) for the customers. BBB offers a road service package as follows: If you pay $ 50, then you will be able to request free road services at most four times in the next 365 days. Now suppose it cost the company BBB $ 100 for each road service. The probability of number of service request is as follows: Number of service requests or up (1) Compute the average cost of road service for BBB in each package contract. (2) Compute the average profit for BBB by selling a road service package. (3) Instead of signing a package contract with BBB, a driver can choose not to participate in any contract. So he has to pay the service fee every time he needs road service. Assume he chooses not to buy the package. Then each time he requests a road service, he has to pay 200 dollars. The probability of the number of requests is the same as the above table. Try to compute his average expense on road service in the next 365 days if he doesn t buy the package. (4) Based on the computation, if he wants to minimize his average expense on road service, should he buy the service package from company BBB?

6 Solution to Homework 8 1. Random variables are (1)(2)(3)(4)(6)(8). 2. E[X ] = 5 and Var[X ] = 1.2. E[Y ] = 1.4 and Var[Y ] = (1) 1/4. (2) 1/13. (3) 12/52 = 3/13. (4) Probabilities are all 1/4. (5) Probabilities are 1/13, 9/13 and 3/13 respectively. (6) Probabilities are 1/13, 3/13, 9/13. (7) 4/13 = (8) Yes. It favors the player. 4. Average is It is against the player. 5. (1) 1/37. (2) Probabilities are 1/37 and 36/37. (3) 1/37. (4) Probabilities: 18/37, 18/37, 1/37. (5) Probabilities: 18/37, 19/37. (6) Expected value: 1/37. (7) Average loss is the same for both games. Both games favor the dealer. 6. E[X ] = and E[Y ] = So EPPLE should choose X as the supplier. 7. (1) Profit of A in a month /3 1/3 Profit B in a month (2) E[A] = 1000 and E[B] = Both stocks have the same average profit. (3) Var[A] = and Var[B] = (4) Buy A because A has small variance. 8. (1) Average cost for BBB is 35. (2) Since a package contract will earn 50 for the company BBB, the average profit will be = 15. (3) Average expense for the customer is 70. (4) He should buy the package.

The student will explain and evaluate the financial impact and consequences of gambling.

What Are the Odds? Standard 12 The student will explain and evaluate the financial impact and consequences of gambling. Lesson Objectives Recognize gambling as a form of risk. Calculate the probabilities