5. Aprimenumberisanumberthatisdivisibleonlyby1anditself. Theprimenumbers less than 100 are listed below.

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1 1. (a) Let x 1,x 2,...,x n be a given data set with mean X. Now let y i = x i + c, for i =1, 2,...,n be a new data set with mean Ȳ,wherecisaconstant. What will be the value of Ȳ compared to X? (b) Let x 1,x 2,...,x n be a given data set with mean X. Now let y i = cx i,fori = 1, 2,...,n be a new data set with mean Ȳ,wherecisaconstant. What will be the value of Ȳ compared to X? (c) Calculate the mean of (a) compare it with the mean of (b) compare it with the mean of , 14.4, Sets A and B are given as follows: A = {1, 5, 7, 8, 10, 13} and B = {1, 2, 4, 7, 10, 15, 1}. Construct the sets A [ B,A\ B,A\ B,B \ A and A 4 B, where4 is called a symmetric di erence and defined as A 4 B =(A \ B) [ (B \ A). A[B = {1, 2, 4, 5, 7, 8, 10, 13, 15, 1},A\B = {1, 7, 10},A\B = {5, 8, 13}, B \ A = {2, 4, 15, 1} and A 4 B = {2, 4, 5, 8, 13, 15, 1}. 3. An experiment consists of rolling two dice. First define its sample space and then write the sets which describe the following events: (a) A: The sum of numbers we get is 7. (b) B: The sum of numbers we get is smaller then 4. (c) C: The sum of numbers we get is larger then 12. (d) D: The number we get for the first die is larger then the number we get for the second die. (e) E: The product of numbers we get is odd. Calculate the probabilities of the events A, B, C, D, E, A [ B,E \ D and B \ E. 1, 1 15, 0,, 1, 1, 1, Three coins are tossed. First define its sample space and then calculate the probability that both heads and tails appear at least once Aprimenumberisanumberthatisdivisibleonlyby1anditself. Theprimenumbers less than 100 are listed below. 1

2 Choose one of these numbers at random. Find the probability that (a) The number is even. (b) The sum of the number s digits is even. (c) The number is greater than , 13 25, In a company the percent of employees who have the VISA credit card is 22%, the percent of employees who have the MASTER credit card is 58%. The percentage of employees who have both cards is 14%. Assume that no other credit cards are possible. What is the percentage of people in this company who don t have any credit card? Asportsclubhas120members,ofwhom44playtennis,30playsquash,and18play both tennis and squash. If a member is chosen at random, find the probability that this person (a) does not play tennis? (b) does not play squash? (c) does not play tennis nor squash? 0.3, 0.75, Astudenthastosell2booksfromhiscollectionofmathematicsbooks,7literature books and 4 economics books. In how many ways can he do that if (a) books must be on the same subject? (b) books must be from di erent subjects? =42, =94. and probabilities are 0.31 and 9. A person has 8 college friends but only 5 of them can be invited to the Black & White Party. (a) In how many ways can this be done if we know that Kate and Serena are not on speaking terms so both of them cannot be invited? 2

3 (b) In how many ways can this be done if we know that Kate and Neal are a couple and will only attend a party if they are both invited? =3, =2andprobabilitiesare0.31 and 0.9. A code consists of a digit chosen from 0 to 9 followed by a letter of the alphabet (A-Z). What is the probability the code is 9Z? We randomly choose a number from the set {1, 2,...,22}. What is the probability that the number is even if we already know that (a) the number is divisible by 3? (b) the number is divisible by 4? (c) the number is divisible by 5? 3 7, 1, Abagcontains3redand4bluemarbles. Twomarblesaredrawnatrandomwithout replacement. If the first marble drawn is blue, what is the probability the second marble is also blue? The coin is tossed, then a die is rolled and then the coin is tossed again. What is the probability that you get the same side of the coin both times and as well that the number you get on the die is divisible by 3? An unfair coin (0.4 probability for heads, 0. probability for tails) is tossed 5 times. (a) What is the probability that you get exactly 3 tails? (b) What is the probability that you get at least 3 tails? 5 3 (0.)3 (0.4) 2, 5 3 (0.)3 (0.4) (0.)4 (0.4) + (0.) In a class of 24 girls, 7 have blonde hair, 5 have red hair and the rest have black hair. Iselectacommitteeof5fromtheclasscompletelyatrandom. Giventhatexactlytwo are red heads, find the probability that my committee contains exactly one blonde. 3

4 Two fair dice are rolled. (a) What is the conditional probability that at least one lands on given that the dice land on di erent numbers? (b) What is the conditional probability that the first one lands on given that the sum of the dice numbers is 7? (c) What is the conditional probability that the first one lands on given that the sum of the dice numbers is 10? (d) What is the conditional probability that the first one lands on given that the sum of the dice numbers is 12? 1 3, 1, 1 3, Let X be a random variable with the following distribution. 2 4 X :. 1/3 1/3 1/3 Calculate EX, Var(X) andsd(x). 4, 2.7, Let X be a random variable with the following distribution X :, a 3/8 1/8 where a 2 R. Find a and calculate EX. a =1/2, EX = Aboxcontains5redand5bluemarbles. Twomarblesarewithdrawnrandomlyat the same time. If they are the same colour, then you win 1.1e. Iftheyareofdi erent colours, then you win -1e. (I.e, you lose 1e). Calculate the expected value of the amount you win e. 4

5 19. It is known that DVDs produced by a certain company will be defective with probability 0.01, independently of each other. The company sells the DVDs in packages of 10 and o ers a money-back guarantee when at least 2 DVDs of the 10 DVDs are defective. What is the probability that company must replace a sold package? 1 (0.99) (0.99) 9 = Let f be defined as follows: where c 2 R. Canfbeadensityfunction? f(x) = c(2x x 3 ): if0<x< 5 2 0: otherwise No, c would have to be 4/225, but then the function f(x) isnolonger non-negative everywhere. For example, look at f(1). 21. Let f be a density function of a random variable X defined as follows: f(x) = a + bx 2 : if 0 apple x apple 1 0: otherwise, where a, b 2 R. IfEX = 3,findaand b. 5 a =3/5,b =/ Let f be a function such that f(x) = (a) Show that f is a density function. 1 x 2 : if x 1 0: otherwise. (b) If X is a random variable with density f, calculateex. (a) (b) EX = Let X be a durability of a light bulb (in hours). The density of X is given by 8 < 2x : if 0 apple x apple f(x) = : if 2 apple x apple 3. : 4 0: otherwise 5

6 (a) Which percent of light bulbs lasts longer than 15 minutes? (b) Calculate EX and Var X. (a) 93.75% = % (b) EX = hours, which is minutes.varx = hours. 24. Let Z be a random variable such that Z N(0, 1). Find the following probabilities: (a) P(Z <0) (b) P(Z 2 > 4) (c) P(Z < 1.75) (d) P( 2 <Z<2) (e) P( Z > 2.8). (a) 1 2 (b) (c) (d) (e) Let Z be a random variable such that Z N(15, 4). Find the following probabilities: (a) P(Z <23) (b) P(Z <11) (c) P(11 <Z<19) (d) P(Z >27). (a) 1 (b) (c) (d) 0

7 2. Let Z be a random variable such that Z N(0, 1). Find the value a, suchthatitholds: (a) P(Z <a)=0.05 (b) P(Z <a)=0.5 (c) P( Z <a)=0.99. (a) 1.4 (b) 0 (c) The time required to complete a certain loan application form is a normal random variable with mean 90 minutes and standard deviation 15 minutes. Find the probability that an application form is filled out in (a) Less than 75 minutes (b) More than 100 minutes (c) Between 90 and 120 minutes. (a) (b) (c) To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed An IQ test produces scores that are normally distributed with mean value 100 and standard deviation The top 1 percent of all scores is in what range? Larger and equal than A producer of cigarettes claims that the mean nicotine content in its cigarettes is 2.4 milligrams with a standard deviation of 0.2 milligrams. Assuming these figures are correct, find the expected value and variance of the sample mean nicotine content of 7

8 (a) 3 (b) 100 (c) 900 randomly chosen cigarettes. (a) X N(2.4, ). E X =2.4, Var X = = (b) X N(2.4, ). E X =2.4, Var X = = (c) X N(2.4, ). E X =2.4, Var X = = Consider a sample of size 1 from a population having mean 100 and standard deviation. Approximate the probability that the sample mean lies between 9 and 104 when (a) =1 (b) =1. Data is assumed to be normal. (a) Approximately 1. (b) Amagazinereportedthat%oftheirreadersreadtheirnewspapersatwork. If300 readers are selected at random, find the probability that exactly 25 say they read the newspaper at work Of the members of a tennis club, 10% are lawyers. If 200 tennis club members are selected at random, find the probability that 10 or more will be lawyers Suppose that we conduct a survey of 19 millionaires to find out what percent of their income the average millionaire donates to charity. We discover that the mean percent is 15. Assume that the standard deviation is 5. Find a 95% confidence interval for the mean percent. Assume that the distribution of all charity percent is approximately normal. 8

9 [12.75, 17.25]. 35. The following data represent the number of drinks sold from a vending machine based on a sample of 20 days: Data is assumed to be normal (a) Determine a 95 percent confidence interval estimate of the mean number of drinks sold daily if the standard deviation is assumed to be 10. (b) Repeat part (a) for the 90 percent confidence interval. (a) [45.22, 53.98] (b) [45.92, 53.28] 3. A survey of 1721 people found that 15.9% of individuals purchase cooking books. Find the 95% confidence interval of the true proportion (expressed in percentages) of people who buy these books. [14.2%, 17.%]. 37. Out of a random sample of 100 students at a university, 82 stated that they were nonsmokers. Based on this, construct a 99 percent confidence interval estimate of the proportion of all the students at the university who are non-smokers. [0.721, 0.919]. 9