INTRODUCTION TO PROBABILITY AND STATISTICS Conditional probability and independent events.
1. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a 1, 2, 3, or 4 on the second toss.
2. Find the probability of not getting a7 or 11 total on either of two tosses of a pair of fair dice.
3. Two cards are drawn from a well-shuffled ordinary deck of52 cards. Find the probability that they are both aces if the first card is (a) replaced, (b) not replaced.
4. Find the probability of a 4 turning up at least once in two tosses of a fair die.
5. Box I contains 3 red and 2 blue marbles while Box II contains 2 red and 8 blue marbles. A fair coin is tossed. If the coin turns up heads, a marble is chosen from Box I; if it turns up tails, a marble is chosen from Box II. Find the probability that a red marble is chosen.
6. Two a’s and two b’s are arranged in an order. All arrangements are equally likely. Given that the last letter, in order, is b, find the probability that the two a’s are together.
Bayes’ theorem.
7. Suppose in Problem 5 that the one who tosses the coin does not reveal whether it has turned up heads or tails (so that the box from which a marble was chosen is not revealed) but does reveal that a red marble was chosen. What is the probability that Box I was chosen (i.e., the coin turned up heads)?
8. In Orange Country,51% of adults are males. One adult randomly selected for a survey involving credit card usage. (a) Find the probability that the selected person is a male. (b) It is later learned that selected survey subject was smoking a cigar. Also 9, 5% of males smoke cigars, whereas1, 7% of females smoke cigars. Find the probability that the selected subject is a male.
9. An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects. (a) If an ELT is randomly selected from the general population of all ELTs, find the probability that it was made by the Altigauge Manufacturing Company. (b) If a randomly selected ELT is then tested and is found to be defective, find the probability that it was made by the Altigauge Manufacturing Company.
10. Suppose that Bob can decide to go to work by one of three modes of transportation, car, bus, or commuter train. Because of high traffic, if he decides to go by car, there is a50% chance he will be late. If he goes by bus, which has special reserved lanes but is sometimes overcrowded, the probability of being late is only 20%. The commuter train is almost never late, with a probability of only 1%, but is more expensive than the bus. (a) Suppose that Bob is late one day, and his boss wishes to estimate the probability that he drove to work that day by car. Since he does not know which mode of transportation Bob usually uses, he gives a prior probability of1/3 to each of the three possibilities. What is the boss’ estimate of the probability that Bob drove to work? (b) Suppose that a coworker of Bob’s knows that he almost always takes the commuter train to work, never takes the bus, but sometimes, 10% of the time, takes the car.
What is the coworkers probability that Bob drove to work that day, given that he was late?
Counting and tree diagrams.
11. Suppose that someone wants to go by bus, by train, or by plane on a week’s vacation to one of the five East North Central States in the USA (Ohio, Indiana, Illinois, Michigan, Wisconsin).
Find the different ways in which this can be done.
12. In how many different ways can one answer all the questions of a true-false test consist of20 questions?
Permutations.
13. How many ways can we rank three people: A, B and C?
14. How many different letter arrangement can be made of the word Columbia?
15. In how many ways can 5 differently colored marbles be arranged in a row?
16. In how many ways can 10 people be seated on a bench if only 4 seats are available?
17. It is required to seat5 men and 4 women in a row so that the women occupy the even places.
How many such arrangements are possible?
18. How many 4-digit numbers can be formed with the 10 digits 0, 1, 2, 3, · · · , 9 if (a) repetitions are allowed, (b) repetitions are not allowed, (c) the last digit must be zero and repetitions are not allowed?
19. Four different mathematics books, six different physics books, and two different chemistry books are to be arranged on a shelf. How many different arrangements are possible if (a) the books in each particular subject must all stand together, (b) only the mathematics books must stand together?
20. In how many ways can7 people be seated at a round table if (a) they can sit anywhere, (b) 2 particular people must not sit next to each other?
Combinations.
21. Out of5 mathematicians and 7 physicists, a committee consisting of 2 mathematicians and 3 physicists is to be formed. In how many ways can this be done if (a) any mathematician and any physicist can be included, (b) one particular physicist must be on the committee, (c) two particular mathematicians cannot be on the committee?
22. From 7 consonants and 5 vowels, how many words can be formed consisting of 4 different consonants and 3 different vowels? The words need not have meaning.
23. How many different salads can be made from lettuce, escarole, endive, watercress, and chicory?
The Binomial coefficients.
24. Find the constant term in the expansion of
1 x + x2
12 . Probability using combinational analysis.
25. A box contains 8 red, 3 white, and 9 blue balls. If 3 balls are drawn at random without replacement, determine the probability that (a) all3 are red, (b) all 3 are white, (c) 2 are red and 1 is white, (d) at least 1 is white, (e) 1 of each color is drawn, (f) the balls are drawn in the order red, white, blue.
26. In the game of poker5 cards are drawn from a pack of 52 well-shuffled cards. Find the proba- bility that (a) 4 are aces, (b) 4 are aces and 1 is a king, (c) 3 are tens and 2 are jacks, (d) a nine, ten, jack, queen, king are obtained in any order, (e) 3 are of any one suit and 2 are of another, (f) at least1 ace is obtained.
Discrete random variables and probability distributions (or probability functions).
27. Suppose that a pair of fair dice are to be tossed, and let the random variable X denote the sum of the points. Obtain the probability distribution for X.
28. Find the probability distribution of boys and girls in families with 3 children, assuming equal probabilities for boys and girls.
29. Find a formula for the probability distribution of the total numbers of heads obtained in four tosses of a balanced coin.
30. Check whether the function given by f(x) = x+ 2
25 for x= 1, 2, 3, 4, 5
can serve as the probability distribution of a discrete random variable.
Discrete (cumulative) distribution functions.
31. (a) Find the distribution function F(x) for the random variable X of Problem 27 and (b) graph this distribution function.
32. Find the distribution function F(x) for the random variable X of Problem 28, and (b) graph this distribution function.
33. If the distribution function of X is given by
F(x) =
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0 for x <2
361 for 2 ≤ x < 3
363 for 3 ≤ x < 4
366 for 4 ≤ x < 5
1036 for 5 ≤ x < 6
1536 for 6 ≤ x < 7
2136 for 7 ≤ x < 8
2636 for 8 ≤ x < 9
3036 for 9 ≤ x < 10
3336 for 10 ≤ x < 11
3536 for 11 ≤ x < 12 1 for x≥ 12 find the probability distribution of this random variable.
Continuous random variables and probability distributions (or probability function or probability density function -pdf-) and (cumulative) distribution functions.
34. If X has the density function f(x) =
ke−3x for x >0
0 for elsewhere find k and P(0.5 ≤ X ≤ 1).
35. Find the distribution function of the random variable X of Problem 34, and use it to reevaluate P(0.5 ≤ X ≤ 1).
36. A random variable X has the density function f(x) = c/(x2 + 1), where −∞ < x < ∞. (a) Find the value of the constant c. (b) Find P(1/√
3 ≤ X ≤ 1).
37. Find the distribution function corresponding to the density function of Problem 36.
38. The distribution function for a random variable X is F(x) =
1 − e−2x for x≥ 0 0 for x <0.
(a) Find the density function. (b) Find the probability that X > 2. (c) Find the probability that−3 < X ≤ 4.
Joint distributions and independent variables.
39. The joint probability function of two discrete random variables X and Y is given by f(x, y) = c(2x + y), where x and y can assume all integers such that 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, and f(x, y) = 0 otherwise. (a) Find the value of the constant c. (b) Find P (X = 2, Y = 1). (c) Find P(X ≥ 1, Y ≤ 2).
40. Find the marginal probability functions (a) of X and (b) of Y for the random variables of Problem 39.
41. Determine the value of k for which the function given by
f(x, y) = kxy for x = 1, 2, 3; y = 1, 2, 3 can serve as a joint probability distribution.
42. Two caplets are selected at random from a bottle containing three aspirin, two sedative and four laxative caplets. If X and Y are, respectively, the numbers of aspirin and sedative caplets included among the two caplets drawn from the bottle, (a) find the probabilities associated with all possible pairs of values of X and Y . (b) Find F(1, 1). (c) Find F (−2, 1). (d) Find F(3.7, 4.5). (e) Find the marginal probability functions of X and Y .
43. The joint density function of two continuous random variables X and Y is
f(x, y) =
cxy 0 < x < 4, 1 < y < 5 0 otherwise.
(a) Find the value of the constant c. (b) Find P(1 < X < 2, 2 < Y < 3). (c) Find P (X ≥ 3, Y ≤ 2).
44. Find the marginal distribution functions (a) of X and (b) of Y for Problem 43.
45. In Problem 43 find P(X + Y < 3).
46. If the joint probability function of X and Y is given by f(x, y) =
x+ y 0 < x < 1, 0 < y < 1 0 elsewhere
find the joint distribution function of these two random variables.
47. Find the joint probability density of the two random variables X and Y whose joint distribution function is given by
F(x, y) =
(1 − e−x)(1 − e−y) x > 0, y > 0
0 elsewhere.
Also, use the joint probability density to determine P(1 < X < 3, 1 < Y < 2).
48. Given the joint probability density
f(x, y) =
2
3(x + 2y) 0 < x < 1, 0 < y < 1
0 elsewhere
find the marginal densities of X and Y .
Independent random variables and Conditional distributions 49. Show that the random variables X and Y of Problem 39 are dependent.
50. Find (a) f(y|2), (b) P (Y = 1|X = 2) for the distribution of Problem 39.
51. If X and Y have the joint density function
f(x, y) =
3
4 + xy, 0 < x < 1, 0 < y < 1, 0, otherwise,
find (a) f(y|x), (b) P
Y > 12 | 12 < X < 12 + dx
52. The joint density function of the random variables X and Y is given by
f(x, y) =
8xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0, otherwise.
Find (a) the marginal density of X, (b) the marginal density of Y , (c) the conditional density of X, (d) the conditional density of Y .
53. Determine whether the random variables of Problem 52 are independent.
Expectation of random variables.
54. In a lottery there are 200 prizes of $5, 20 prizes of $25, and 5 prizes of $100. Assuming that 10, 000 tickets are to be issued and sold, what is a fair price to pay for a ticket?
55. Find the expectation of a discrete random variable X whose probability function is given by f(x) =
1 2
x
, x= 1, 2, 3, · · · .
56. Find the expectation of the sum of points in tossing a pair of fair dice.
57. A continuous random variable X has probability density given by
f(x) =
2e−2x, x >0,
0, x≤ 0.
Find (a) E(X), (b) E(X2).
58. The joint density function of two random variables X and Y is given by f(x, y) =
xy/96, 0 < x < 4, 1 < y < 5, 0, otherwise.
Find (a) E(X), (b) E(Y ), (c) E(XY ), (d) E(2X + 3Y ).
59. If X is the number of points rolled with a balanced die, find the expected value of g(X) = 2X2+ 1.
60. If X has the probability density
f(x) =
e−x, x >0, 0, elsewhere.
Find the expected value of g(X) = e3X/4.
61. With reference to Problem 42, find the expected value of g(X, Y ) = X + Y . 62. If the joint probability density of X and Y is given by
f(x, y) =
2
7(x + 2y), 0 < x < 1, 1 < y < 2,
0, otherwise.
Find the expected value of g(X, Y ) = X/Y3. Variance and standard deviation.
63. Find (a) the variance, (b) the standard deviation of the sum obtained in tossing a pair of fair dice.
64. Find (a) the variance, (b) the standard deviation for the random variable of Problem 57.
65. If X represents the number of heads that appear when one coin is tossed and Y the number of heads that appear when two coins are tossed. Compare the variance of the random variables X and Y .
Moments and moment generating functions.
66. Given that X has the probability distribution f(x) = 18 3
x
for x= 0, 1, 2, 3. Find the moment generating function of this random variable and use it to determine µ1 and µ2.
67. The random variable X can assume the values 1 and −1 with probability 1/2 each. Find (a) the moment generating function, (b) the first four moments about the origin.
68. A random variable X has density function given by
f(x) =
2e−2x, x≥ 0, 0, x <0.
Find (a) the moment generating function, (b) the first four moments about the origin.
69. Find the first four moments (a) about the origin, (b) about the mean, for a random variable X having density function
f(x) =
4x(9 − x2)/81, 0 ≤ x ≤ 3,
0, otherwise.
70. Find the moment generating function of the random variable whose probability density is given by
f(x) =
e−x, x >0, 0, elsewhere, and use it to find an expression for µr.
Covariance and correlation coefficient.
71. Find (a) E(X), (b) E(Y ), (c) E(XY ), (d) E(X2), (e) E(Y2), (f) V ar(X), (g) V ar(Y ), (h) Cov(X, Y ) if the random variables X and Y are defined as in Problem 39.
72. Work Problem 71 if the random variables X and Y are defined as f(x, y) =
1
210(2x + y), 2 < x < 6, 0 < y < 5
0, otherwise.
73. Find the covariance of the random variables X and Y whose joint probability density is given by
f(x, y) =
2, x > 0, y > 0, x + y < 1, 0, otherwise.
74. If the joint probability distribution of X and Y is given by X
-1 0 1
-1 16 13 16 23
Y 0 0 0 0 0
1 16 0 16 13
13 1
3 1
3 1
show that their covariance is zero even though the two random variables are not independent.
75. Find the correlation coefficients of Problem 71 and Problem 72.
Conditional expectation, variance, and moments.
76. Find the conditional expectation of Y given X= 2 in Problem 39.
77. Find the conditional expectation of (a) Y given X, (b) X given Y in Problem 52.
78. Find the conditional variance of Y given X for Problem 52.
79. With reference to Problem 42, find the conditional mean of X given Y = 1.
80. If the joint probability density of X and Y is given by
f(x, y) =
2
3(x + 2y), for 0 < x < 1, 0 < y < 1.
0, elsewhere,
find the conditional mean and the conditional variance of X given Y = 1/2.
81. Little Mac is boxing for glory and prize money. In his first match he faces Super Macho Man. If Little Mac wins, then he faces Mike Tyson in a championship bout. If Little Mac loses, then he faces Glass Joe in a consolation match. The prize money is$10, 000 for winning the champions- hip bout,$1, 000 for loosing the championship bout, $100 for winning the consolation match, and $0 loosing the consolation match. Little Mac has a 50% chance of beating Super Macho Man, only a 10% chance of beating Tyson, and a 90% chance of beating Glass Joe. (a) Draw the event tree for this situation. (b) What is the expectation of the prize money, conditional on winning the first match? (c) What is the expectation of the prize money, conditional on losing the first match? (d) What is the expectation of the prize money before the result of the first match is known?
Chebyshev’s inequality.
82. For the random variable of Problem 68, (a) find P(|X −µ| ≥ 1). (b) Use Chebyshev’s inequality to obtain an upper bound on P(|X − µ| ≥ 1) and compare with the result in (a).
The binomial distribution.
83. Find the probability of getting more than one head in 5 tosses of a fair coin.
84. Find the probability that in tossing a fair coin three times, there will appear (a)3 heads, (b) 2 tails and 1 head, (c) at least 1 head, (d) not more than 1 tail.
85. Find the probability that in five tosses of a fair die, a3 will appear (a) twice, (b) at most once, (c) at least two times.
86. Find the probability that in a family of 4 children there will be (a) at least 1 boy, (b) at least 1 boy and at least 1 girl. Assume that the probability of a male birth is 1/2.
87. Out of2000 families with 4 children each, how many would you expect to have (a) at least 1 boy, (b)2 boys, (c) 1 or 2 girls, (d) no girls?
88. If20% of the bolts produced by a machine are defective, determine the probability that out of 4 bolts chosen at random, (a) 1, (b) 0, (c) less than 2, bolts will be defective.
89. If the probability of a defective bolt is 0.1, find (a) the mean, (b) the standard deviation, for the number of defective bolts in a total of400 bolts.
The normal distribution.
90. Find the probabilities that a random variable having the standard normal distribution will take on a value (a) less than 1.72, (b) less that −0.88, (c) between 1.30 to 1.75, (c) between
−0.25 and 0.45.
91. If “area” refers to that under the standard normal curve, find the value or values of z such that (a) area between 0 and z is 0.3770, (b) area to left of z is 0.8621, (c) area between −1.5 and z is0.0217.
92. The mean weight of500 male students at a certain college is 151 lb and the standard deviation is 15 lb. Assuming that the weights are normally distributed, find how many students weigh (a) between 120 and 155 lb, (b) more than 185 lb.
93. The mean inside diameter of a sample of 200 washers produced by a machine is 0.502 inches and the standard deviation is0.005 inches. The purpose for which these washers are intended allows a maximum tolerance in the diameter of0.496 to 0.508 inches, otherwise the washers are considered defective. Determine the percentage of defective washers produced by the machine, assuming the diameters are normally distributed.
Normal approximation to binomial distribution.
94. Find the probability of getting between 3 and 6 heads inclusive in 10 tosses of a fair coin by using (a) the binomial distribution, (b) the normal approximation to the binomial distribution.
95. A fair coin is tossed 500 times. Find the probability that the number of heads will not differ from250 by (a) more than 10, (b) more than 30.
96. A die is tossed120 times. Find the probability that the face 4 will turn up (a) 18 times or less, (b)14 times or less, assuming the die is fair.
Poisson distribution.
97. The average number of homes sold by the Acme Realty company is2 homes per day. What is the probability that exactly3 homes will be sold tomorrow?
98. On an average Friday, a waitress gets no tip from 5 customers. Find the probability that she will get no tip from7 customers this Friday.
99. During a typical football game, a coach can expect3.2 injuries. Find the probability that the team should have at most1 injury in this game.
Poisson approximation to binomial distribution.
100. Ten percent of the tools produced in a certain manufacturing process turn out to be defective.
Find the probability that in a sample of 10 tools chosen at random, exactly 2 will be defec- tive, by using (a) the binomial distribution, (b) the Poisson approximation to the binomial distribution.
101. A computer chip contains 1000 transistors. Each transistor has probability 0.0025 of being defective. What is the probability that the chip contains at most 4 defective transistors?
