seven with type A defects, five with type B defects, and two with both types of defects. Let S be the sample space generated by the operation of drawing one item blindly from the box. Let be the event of getting a type A defective and the event of getting a type B defective. Set up a Venn diagram and show the following events: indicating the number of elements in each. How many elements are in ?In the box described in part (a) of this problem, let be the sample space generated by blindly drawing two items simultaneously from the box. Let G be the event that corresponds to pairs of nondefective items. Let be the event that corresponds to pairs of items containing at least one type A defective while is the event that corresponds to pairs of items containing at least one type B defective. Set up a Venn diagram and show the eight basic events in obtained by placing or omitting a bar over each label in , and write in the number of elements in each of the eight basic events.4 A “true” icosahedral (Japanese) die has 20 sides, two sides marked with 0, two sides with 1, two sides with , two sides with 9. The probabilities assigned to the 20 faces are all equal. Suppose then that three such dice are thrown. Find the following probabilities:That no two top faces will be alike.That at least two top faces will be alike.That all three top faces will be different even numbers (0 is considered an even number).Generalize parts (a) and (b) to the case of a true ‐sided die with two sides marked 1, two sides marked , two sides marked n.
5 Ten defective items are known to be in a box of 100 items.If they are located by testing the items one at a time until all defectives are found, what is the probability that the 10th (last) defective item is located when the 50th item is tested?What is the probability that if 50 items are drawn at random from the box and tested, all 10 defectives will be found?If 20 are tested and found to be nondefective, what is the probability that all defectives will be found among the next 30 tested?
6 If a lot of 1000 articles has 100 defectives and if a sample of 10 articles is selected at random from the lot, what is the probability that the sample will contain:No defectives?At least one defective?
7 Assume that a given type of aircraft motor will operate eight hours without failure with probability 0.99. Assume that a two‐motor plane can fly with at least one motor, that a four‐motor plane can fly with at least two motors, and that failure of one motor is independent of the failure of another.If a two‐motor plane and a four‐motor plane take off for an eight hour flight, show that the two‐motor plane is more than 25 times more likely to be forced down by motor failure than the four‐motor plane.Compute the respective probabilities that the planes will not be forced down by motor failure.What is the answer to (b) if the probability of failure of a motor during an 8‐hour period is p rather than 0.01?
8 Suppose 10 chips are marked , respectively, and put in a hat. If two chips are simultaneously drawn at random, what is the probability thatTheir difference will be exactly 1?Neither number will exceed 5?Both numbers will be even?At least one of the numbers will be 1 or 10?
9 If the probability is 0.001 that a type‐X 20‐W bulb will fail in a 10‐hour test, what is the probability that a sign constructed from 1000 such bulbs will burn 10 hours:With no bulb failures?With one bulb failure?With k bulb failures?
10 The game of craps is played with two ordinary six‐sided dice as follows: If the shooter throws 7 or 11, he wins without further throwing; if he throws 2, 3, or 12, he loses without further throwing. If he throws 4, 5, 6, 8, 9, or 10, he must continue throwing until a 7 or the “point” he initially threw appears. If after continuing a 7 appears first he loses; if the “point” he initially threw appears first, he wins. Show that the probability is approximately 0.4929 that the shooter wins (assuming true dice).
11 In a group of 11 persons, no two persons are of the same age. We are to choose five people at random from this group of 11.What is the probability that the oldest and the youngest persons of the 11 will be among those chosen?What is the probability that the third youngest of the 5 chosen will be the 6th youngest of the 11?What is the probability that at least three of four of the youngest of the 11 will be chosen?
12 Suppose that the probability is 1/365 that a person selected at random was born on any specified day of the year (ignoring persons born on February 29). What is the probability that if r people are randomly selected, no two will have the same birthday? (The smallest value of r for which the probability that at least two will have a common birthday exceeds 0.5 is 23.)
13 Suppose that six true dice are rolled simultaneously. What is the probability of gettingAll faces alike?No two faces alike?Only five different faces?
14 , , , and are four events that are such that, , . Express the values of the following probabilities in terms of :Probability of the occurrence of exactly 1, exactly 2, exactly 3, of the events .
15 If four addressed letters are inserted into four addressed envelopes at random, what is the probability thatNo letter is inserted into its own envelope?At least one letter is inserted into its own envelope?
16 Three machines A, B, and C produce 40%, 45%, and 15%, respectively, of the total number of nuts produced by a certain factory. The percentages of defective output of these machines are 3%, 6%, and 9%. If a nut is selected at random, find the probability that the item is defective.
17 In Problem 16, suppose that a nut is selected at random and is found to be defective. Find the probability that the item was produced by machine A.
18 Enrollment data at a certain college shows that 30% of the men and 10% of the women are studying statistics and that the men form 45% of the student body. If a student is selected at random and is found to be studying statistics, determine the probability that the student is a woman.
19 A certain cancer diagnostic test is 95% accurate on those that do have cancer, and 90% accurate on those that do not have cancer. If 0.5% of the population actually does have cancer, compute the probability that a particular individual has cancer if the test finds that he has cancer.
20 An urn contains three blue and seven white chips. A chip is selected at random. If the color of the chip selected is white, it is replaced and two more white chips are added to the urn. However, if the chip drawn is blue, it is not replaced and no additional chips are put in the urn. A chip is then drawn from the urn a second time. What is the probability that it is white?
21 Referring to Problem 20, suppose that we are given that the chip selected for the second time is white. What is the probability that the chip selected at the first stage is blue?
22 Referring to Problem 5, suppose that it takes 11 tests to find all the 10 defectives, that is, the 11th test produces the last defective. What is the probability that the first item is nondefective?
23 A bag contains a nickel, quarter, and “dime”, with the dime being a fake coin and having two heads. A coin is chosen at random from the bag and tossed four times in succession. If the result is four heads, what is the probability that the fake dime has been used?
24 In a playground, there are 18 players, 11 of them boys and seven are girls. Eight of the boys and three of the girls are soccer players; the rest are basketball players. The name of each player is written on a separate slip, and then these slips are put into an urn. One slip is drawn randomly from the urn; the player whose name appears on this slip is given a prize. What is the probability that a soccer player gets the prize given that a boy gets the prize?
25 Let , and be mutually exclusive and exhaustive events in a sample space S, and let , and . Let B be another event in S such that . Find the probabilities and .
26 An industry uses three methods , and to train their workers. Of all the workers trained, 50% are trained by method , 28% by method , and the rest, 22%, by method . Further 10% of those trained by method do not perform their job well, while 5% trained by method and 15% by methods also do not perform their job well. A randomly
