Math 4610, Problems to be Worked in Class

Transcription

1 Math 4610, Problems to be Worked in Class Bring this handout to class always! You will need it. If you wish to use an expanded version of this handout with space to write solutions, you can download one chapter by chapter on my web site: Note on numbering: Each problem is numbered as <Chapter#.Section#.Problem# >. The rst two numbers always refer to the relevant chapter and section in the textbook. The third number is used only for internal organization of this handout. Chapter 1 Problem Two fair dice are rolled. a) Find the chance that both dice show an odd number. b) Find the chance that the sum of the numbers on the dice is 8. Problem What is the distribution of the length of a word, picked at random from the sentence \A quick brown fox jumps over the lazy dog"? Problem A box contains three two-sided cards. The rst card is black on both sides, the second is white on both sides, and the third is black on one side, and white on the other. A card is drawn at random from the box and placed randomly on the table. If the visible side of the card is black, what is the chance that the other side is white? Problem Two cards are dealt from a well-shued standard deck of 52 cards. Find: a) the chance that the rst card is a king b) the chance that the second card is a king c) the chance that both cards are kings. Problem An electrical device consists of two components. The rst component can fail with a probability of 15%. If the rst component fails, the second component will fail also with a probability of 20$. If the rst component does not fail, the second component will fail with a probability of only 10%. Find: a) the chance that the second component works (i.e. does not fail) b) the chance that one component works and the other fails. Problem A hat contains three coins. The rst coin is fair, the second lands heads one-third of the time, and the third lands heads two-fths of the time. A coin is picked at random from the hat and ipped. What is the chance the coin lands heads? 1

2 Problem Two electrical components perform independently of each other. The rst works with probability 0:4 and fails with probability 0:6, the second works with probability 0:5 and fails with probability 0:5. Find the chance that: a) both components fail b) exactly one component works Problem A blood test is used to test for a certain disease. Suppose 95% of people who have the disease test positive. On the other hand, 1% of people without the disease test positive too (false positives). Suppose rther that 0:5% of the population has the disease. a) A person is chosen at random and tests positive. Given this, what is the probability that the person has the disease? b) Answer the same question, but this time given that the test was negative. Problem (Guessing the box) There are three boxes: Box 1 contains one white and one black ball; Box 2 contains two white balls and one black ball; and Box 3 contains three white balls and one black ball. A box is picked at random, and a ball is pulled at random from that box. For i = 1; 2; 3, nd the probability that the box selected was Box i, given that the ball is white. Problem (Continuation of Problem ) A patient enters the doctor's oce, feeling ill. Taking into account the patient's symptoms, age, gender, and family history, the doctor estimates the probability that the patient has the disease to be 25%. If the blood test turns out positive, how should the doctor revise his opinion? Problem A ve-card hand is dealt from a deck of 52. What is the probability that the hand is a ush? (i.e. ve cards from the same suit) Problem (Birthday Problem) There are n people at a party. What is the probability that at least two of them have the same birthday? Problem Suppose each of the switches in the following circuit is closed with probability p i, and open with probability q i = 1 p i, i = 1; : : : ; 5. Assuming that the switches act independently, calculate the probability that current will ow through the circuit. (Figure from p.68 of the textbook goes here) 2

3 Problem Consider a sequence of independent Bernoulli(p) trials. a) What is the probability that exactly k trials are needed to get a success? b) What is the probability that at most k trials are needed to get a success? Problem You roll a fair die repeatedly until it shows the number 6. Find the chance that this takes: a) at most 4 rolls b) exactly 4 rolls c) more than 4 rolls Chapter 2 Problem A man res 8 shots at a target. Assume that the shots are independent, and each shot hits the bull's eye with probability 0:7. a) What is the chance that he hits the bull's eye exactly 4 times? b) Given that he hit the bull's eye at least twice, what is the chance that he hit the bull's eye exactly 4 times? c) Given that the rst two shots hit the bull's eye, what is the chance that he hits the bull's eye exactly 4 times in the 8 shots? Problem A survey organization takes a random sample of 200 voters from a district. If 45% of the voters in the district oppose a certain ballot measure, estimate the chance that: a) exactly 90 voters in the sample oppose the measure b) more than half the voters in the sample oppose the measure Problem Two fair dice are rolled 60 times. Call each time the dice are rolled a trial. Find the chance that double-six happens exactly twice in the 60 trials: a) exactly (using the binomial distribution formula) b) using the normal approximation c) using the Poisson approximation Problem Repeat Problem , but now for 600 trials. Problem A company produces plasma TVs, 99% of which work properly, and 1% of which are defective. A sample of 200 TVs is taken. What is the chance that the sample contains at least 198 TVs that work properly? Problem A deck of cards is shued and dealt to four players, with each receiving 13 cards. Find: a) the probability that the rst player holds all the aces b) the probability that the rst player holds all the aces given that she holds the ace of hearts c) the probability that the rst player holds all the aces given that she holds at least one d) the probability that the second player holds all the aces given that he holds all the hearts 3

4 Chapter 3 Problem Let X have a uniform distribution on the integers 10; 9; : : : ; 9; 10. That is, P (X = x) = 1 for x = 10; 9; : : : ; 9; 10. Find: 21 a) P (3X 7) b) P (X 2 7) c) P ( p X + 10 > 4) d) P (e X 1) e) P (jx 4j > 3) Problem Two draws are made without replacement from a box containing three tickets, labeled 1, 2 and 3. Let X denote the rst number drawn, and Y the second number. a) Find the joint distribution of X and Y. b) Find the distribution of Z := X Y. Problem Three numbers are picked at random, without replacement, from the set f0; 1; : : : ; 9g. Let X be the smallest, and Y the largest of the three numbers drawn. a) Find the joint distribution of X and Y. b) Find the distribution of S := X + Y. Problem You roll a fair die 10 times. What is the chance that you get exactly 3 ves, 2 sixes, and 5 others? Problem (Independent or not?) A box with 10 tickets contains some number r of red tickets, and 10 r green tickets. A sample of size 100 is drawn at random from the box with replacement. Then a second sample of size 100 is drawn with replacement. Let X i denote the number of red tickets in the ith sample (i = 1; 2). Are X 1 and X 2 independent? Explain! Problem Let X have the uniform distribution on f1; 2; 3g, and let Y = X 2. a) Find the distribution of Y. b) Find E(Y ) = E(X 2 ), and compare with [E(X)] 2. Problem A fair coin is tossed twice. Let X be the number of heads, and Y the number of tails in the two tosses. Show that E(XY ) 6= E(X)E(Y ). Problem Ten dice are rolled. Find the expectation of the sum of the dice. Problem A building has 10 oors above the basement. If 12 people get into an elevator at the basement, and each chooses a oor at random to get out, independently of the others, at how many oors do you expect the elevator to make a stop to let out one or more of these 12 people? 4

5 Problem Three dice are rolled. Let M be the minimum of the three numbers rolled, and let S be the sum of the three numbers rolled. a) Find E(M). b) Find E(S). Problem The 13 spades of a deck of cards are dealt one by one. Let X be the number of cards before the ace, and Y the number of cards after the ace. a) Show that SD(X) = SD(Y ). b) Find E(X) and E(Y ). Problem Suppose the average family income in an area is $10,000. a) Find an upper bound for the percentage of families with incomes over $50,000. b) Find a better upper bound if it is known that the standard deviation of incomes is $8000. Problem A die is rolled repeatedly until 6 appears. Find the probability that the die is rolled a) an even number of times b) an odd number of times Problem Two players, A and B, play a sequence of independent games. Each game is won by A with probability P (A), won by B with probability P (B), or drawn with probability P (D). Suppose they play until the rst decisive game, and call the winner of that game the overall winner. Find the probability that A is the overall winner. Problem Let X denote the number of times you need to roll a die until 6 has appeared three times. What is the distribution of X? Problem How many raisins must cookies contain on average for the chance of a cookie containing at least one raisin to be at least 99%? Problem Let X P s(1), Y P s(2), and assume X and Y are independent. Find the probability that the average of X and Y equals 5. Chapter 4 Problem Measurements on the weight of a lump of metal are believed to be independent and identically distributed; each measurement has mean 12 grams and SD 1.1 gram. a) Find the chance that a single measurement is between 11.8 and 12.2 grams, assuming that individual measurements are normally distributed. b) Estimate the chance that the average of 100 measurements is between 11.8 and 12.2 grams. For this calculation, is it necessary to assume that individual measurements are normally distributed? Explain. Problem Suppose the lifetime of a light bulb has an exponential distribution with a mean of 800 hours. What is the chance that it will last more than 1000 hours? 5

6 Problem Suppose the time you have to wait for the bus has an exponential distribution with a mean of 15 minutes. If you have already waited 10 minutes, what is the chance the bus will arrive in the next 5 minutes? Problem A particular kind of atom has a half-life of 1 year. Find: a) the probability that an atom of this type survives at least 5 years b) the time at which the expected number of atoms is 10% of the original c) if there are 1024 atoms present initially, the time at which the expected number of atoms remaining is one d) the chance that in fact none of the 1024 original atoms remains after the time calculated in c) Problem Let X be a N(; 2 ) random variable, and let Y = e X. a) Find the density of Y. b) Find the mean and variance of Y. (They are not and 2!) 6