MIDTERM PRACTICE PROBLEMS PSTAT 120A - F17 Professors Chen and Myung 1. Beyonce is having twins! Twins come in two different types: identical or fraternal. Identical twins are alike genetically (including gender) and are typically very similar in appearance. Fraternal twins are genetically distinct and are not necessarily of the same gender. Suppose there is a 30% chance the Beyonce will have identical twins and that each child is a boy or a girl with equal probability, independently. (a) Find the probability that Beyonce has one boy and one girl. (b) Given that Beyonce has two boys, what is the probability that they are fraternal twins. 2. President Drumpf can't decide how many cabinet members to fire today. To decide, he throws a six-sided die twice and sums up the numbers on the two rolls. He will then fire that number of cabinet members. However, he uses his own presidential \"You're Fired!\" dice where each die has the following: three sides with the number 0, two sides with the number 1, and one side with the number 3. Let X denote the sum of the numbers on the two dice. (a) Find the state space of X. (b) Find the probability mass function of X. (c) Compute FX p5.5q. 3. Alternative Game of Thrones facts have been discovered in the archives! Jon Snow's campaign to Make The Wall Great Again involves teaching 1000 wildlings how to wield a dragonglass (obsidian) sword. (a) Jon wants to have at least 3 wildling guards who know how to wield a dragonglass sword. If each wildling can independently master the sword with probability 0.002, find the probability that Jon will have his guards. (b) Approximate the probability you found in part (a) with a Poisson distribution. Do not leave your answer in an infinite sum. (c) Is using a Poisson distribution for an approximation a good choice in this case? Explain your answer in a few brief sentences. 4. A Halloween trick-or-treat group consists of 3 Trolls, 4 Nazg uls, 4 Ents, and 5 Pokemon. A committee of 4 is to be picked to represent the group at the Monster Bash. (a) Find the probability that the committee will consist of 1 from each type of monster. (b) Suppose that the Pokemon refuse to be on the same committee as a the Trolls. Find the probability that this type of committee is formed. 5. Your professor likes to exclusively buy boxes of cereal with prizes inside. However, she has found that in fact only about 90% of the boxes of cereal that she's bought in the past have prizes in them. If your professor lives off of cereal and buys 50 boxes of cereal over the next year, (a) find the probability that she will collect more than 47 prizes. (b) Approximate the probability you found in part (a) with a Poisson distribution. Do not leave your answer in an infinite sum. (c) Is using a Poisson distribution for an approximation a good choice in this case? Explain your answer in a few brief sentences. 6. Rey is trying to escape Kylo Ren (and the dark side). Rey is equally likely to choose one of the three vehicles to attempt her escape: the Millennium Falcon, a stolen TIE fighter, and a X-Wing Starfighter. If she pilots the Millennium Falcon, she has a 0.4 probability of escaping; if she pilots a stolen TIE fighter, she has a 0.6 probability of escaping; and if she pilots a X-Wing Starfighter, she has a 0.7 probability of escaping. Given that she escapes, what is the probability that she piloted the Millennium Falcon? 7. An urn initially contains 5 red marbles and 7 blue marbles. Bored out of your mind, you decide to play a \"game\" that goes as follows. During each round, you randomly pull a marble out of the urn. If the marble you chose was red, you return the marble back into the urn along with 1 more blue marble. If the marble you chose was blue, you put the marble back into the urn along with 2 more red marbles. What is the probability that on the first round you drew a red marble and on the third round you drew a blue marble? 8. After watching Guardians of the Galaxy, our class became extremely motivated to befriend a raccoon. In this class there are 20 males and 16 females. (a) We are going to make two committees from the people in this class. The first committee will consist of 6 people, 3 males and 3 females, to search for a raccoon to befriend; the second committee will consist of 4 other people, 2 males and 2 females, to wait near a phone and medical supplies so that after the first committee actually finds a raccoon and realizes that they're vicious beasts, they can be bandaged up and an ambulance can be called. How many ways are there for us to create these committees? (b) After everyone survived the \"raccoon incident\" (and I lost my job), all 36 people in this class return to the classroom and shake everyone else's hands (no repeated handshakes) on a job well done. How many handshakes took place? 9. There are three urns: Urn A, Urn B, and Urn C. Urn A contains 4 red marbles and 6 blue marbles; Urn B contains 3 red marbles and 7 blue marbles; Urn C has 8 red marbles and 2 blue marbles. You are going to select a random marble from one of the urns; there is a 1{2 probability you will draw the marble from Urn A; there is a 3{10 probabilty you will draw a marble from Urn B; there is a 1{5 probability you will draw the marble from Urn C. (a) Let R be the event that you select a red marble. Find P pRq. (b) Given that the marble you draw is red, what is the probability that you drew the marble from Urn A? 10. How many distinguishable letter arrangements can be made from the word BEYONCE so that the E s are not next to each other? 11. An alien classroom has 29 students: 17 glargons and 12 bubuus. A committee of 2 students is to be selected. Page 2 (a) For k \" 0, 1, or 2, what is the number of ways to choose this committee if k of the students are to be bubuus? Note: You can either write out for each case k \" 0, k \" 1, and k \" 2 explicitly, or you can instead just write one general formula in terms of k. (b) Suppose that each possible committee is equally likely. Let X be (the random variable giving) the number of bubuus selected for the committee. Find the probability mass function pX of X. 12. Little Red is trying to get to her grandmother's house, but the Wolf is trying to stop her! To get to her grandmother's house, Little Red walks down a road until the road splits. The split forces her to either go left or right. If she goes to the left she must independently find either a cloak of invisibility or a sword (or both) to get past the Wolf and get to her grandmother's house. If she goes right she must find the Wolf-Off! spray to get past the Wolf and get to her grandmother's house. Assume that Little Red has a probability pL of going left at the split in the road, and pR of going right. Given that Little Red goes left at the split, she has a probability pC to find the cloak of invisibility and probability pS to find the sword. Given that she goes right at the split, then she has a probability of pW of finding the Wolf-Off! spray. Question: What is the probability that Little Red makes it to her grandmother's house? 13. Suppose you shuffle together two standard 52 card decks (effectively, you double the number of each card). You are randomly dealt a 5 card poker hand. Let X be the random variable counting the number of 2's in your hand. (a) What is the state space SX of X? (b) What is the probability mass function pX of X? That is, find pX pkq \" P pX \" kq for k in SX . (c) What is the value of the distribution FX of X at 5.1? That is, find FX p5.1q \" P pX 5.1q. 14. Your jar of Bertie Bott's Every Flavour Beans contains only 2 flavors: 8 flavored like Strawberry and Peanut-Butter Ice Cream (SPB) and 4 flavored like Lobster (L). You play the following game with your friend James (who loves SPB): If James draw a SPB, he eats it and you add 3 L to the jar. If James draws an L, you eat it and you add 1 SPB to the jar. (a) What is the probability that James draws 3 SPB in a row? (b) What is the probability that on James' second draw, he draws L? 15. At the fair, you find a carnival game that interests you. It is a dart and balloon game (sometimes called Balloon Darts) with 6 balloons: 3 green, 2 blue, 1 red. During each round, you throw 3 darts and pop 3 balloons. Each popped green balloon is worth 2 points; each blue balloon is worth 3 points; and the red balloon is worth 4 points. You decide to play. (a) Given this \"experiment\" of popping three balloons, what is a reasonable sample space? (b) Let X be the (random variable representing the) number of points earned during one round of the game. What is the state space of X? (c) Let X be the random variable from the previous part and assume that each balloon is equally likely to be popped. Find the probability mass function of X. (d) Compute FX p11.2q. Page 3 (e) If during a round you gain at least 10 points, you will win a bowl of Bacon-Jalape no-Macand-Cheese; otherwise, you are awarded a glass of water. Assume that the outcome of consecutive rounds are independent and that you can not carry points between rounds. What is the probability that within 5 rounds of play, you win at least 2 bowls of BaconJalape no-Mac-and-Cheese? 16. At the Big E, there are houses representing the states Massachusetts, Maine, and Connecticut. Assume that your friend Sasha is equally likely to go into exactly one of these houses, though you do not know which one. Sasha will eat either an apple cider doughnut or a lobster roll, but not both. You figure that if she went into the Massachusetts house, she has a 2{3 probability of getting an apple cider doughnut and a 1{3 probability of getting a lobster roll; if she went into the Maine house, she has a 1{4 probability of getting an apple cider doughnut and a 3{4 probability of getting a lobster roll; if she went into the Connecticut house, she has a 5{6 probability of getting an apple cider doughnut and a 1{6 probability of getting a lobster roll. Given that Sasha is eating a lobster roll, what is the probability that she went into the Massachusetts house? 17. Consider a game at the Big E where the dealer rolls two fair six-sided dice. Let F be the event that one of the two dice rolled is a 4. Let S be the event that the sum of the two rolls is 7. Check to see if these events are independent. (Whether or not you judge them to be independent, you must show your work to prove your case.) 18. Tired of lightsaber battles, you decide to play a new game. The game goes as follows: Initially there are two marbles in an urn; one marble is red, the other is green. At each round of the game, you will randomly select a marble, then put the marble back into the urn along with another marble of the opposite color (e.g., if you draw a red marble, you will put the red marble back into the urn along with a green marble). The game is over when either you draw a green marble, or you have played for three rounds without drawing a green marble (whichever happens first). (a) Give a reasonable sample space for this game. (b) Let X be the random variable representing the number of rounds you play until the game ends. What is the state space of X? (c) What is the probability mass function of X? (d) What is the expected value of X? (e) Compute the variance of X. 19. To reconnect and revive the Darth Vader fanbase, Kylo Ren opens a Snapchat account. He's creating a screen name using the letters DARKSIDEV ADER (a) How many screen names (distinguishable letter arrangements) are possible? (b) Let V be the event that the letters V ADER are together in that order. What is P pV q? 20. Consider the following graph of the distribution FX ptq of X Page 4 defined by $ ' 0 t0 ' ' ' &.3 0 t 1 FX ptq \" '.8 1 t 3 ' ' ' %1 t 3 (a) Is the random variable X discrete, continuous, or neither? (b) What is the state space SX of X? (c) What is the expected value ErXs? Page 5