Elementary Probability For Applications 1st Edition Rick Durrett - Solutions

In the freshman class, 62% of the students take math, 49% take science, and 38% take both science and math. What percentage takes at least one science or math course?
24% of people have American Express cards, 61% have Visa cards, and 8% have both. What percentage of people have at least one credit card?
Suppose Ω = {a, b, c }, P ({a, b}) = 0.7, and P ({b, c }) = 0.6. Compute the probabilities of {a}, {b}, and {c}.
Suppose A and B are disjoint with P (A) = 0.3 and P (B) = 0.5. What is P (Ac ∩ Bc )?
Given two events A and B with P (A) = 0.4 and P (B) = 0.7, what are the maximum and minimum possible values for P (A ∩ B)?
Suppose we draw 2 cards out of a deck of 52. Let A = “the first card is an ace” and B = “the second card is a spade.” Are A and B independent?
A family has 3 children, each of whom is a boy or a girl with probability 1/2.Let A = “there is at most 1 girl” B = “the family has children of both sexes.”(a) Are A and B independent? (b) Are A and B independent if the family has 4 children?
Suppose we roll a red and a green die. Let A = “the red die shows a 2 or a 5” and B = “the sum of the two dice is at least 7. ” Are A and B independent?
Roll two dice. Let A = “the sum is even” and B = “the sum is divisible by 3,” that is, B = {3, 6, 9, 12}. Are A and B independent?
Roll two dice. Let A = “the first die is odd,” B = “the second die is odd,” and C = “the sum is odd.” Show that these events are pairwise independent but not independent.
Nine children are seated at random in three rows of three desks. Let A = “Al and Bobby sit in the same row” and B = “Al and Bobby both sit at one of the four corner desks.” Are A and B independent?
Two students, Alice and Betty, are registered for a statistics class. Alice attends 80% of the time, Betty 60% of the time, and their absences are independent. On a given day, what is the probability that(a) At least one of these students is in class and(b) Exactly one of them is there?
Let A and B be two independent events with P (A) = 0.4 and P (A ∪ B) = 0.64.What is P (B)?
Three students each have probability 1/3 of solving a problem. What is the probability that at least one of them will solve the problem?
Three independent events have probabilities 1/4, 1/3, and 1/2. What is the probability that exactly one will occur?
Three missiles are fired at a target. They will hit it with probabilities 0.2, 0.4, and 0.6. Find the probability that the target is hit by(a) Three.(b) Two.(c) One.(d) No missiles.
Three couples that were invited to dinner will independently show up with probabilities 0.9, 0.89, and 0.75. Let N be the number of couples that show up. Calculate the probability P (N) with N = 3, 2, 1, 0.
A college student takes 4 courses a semester for 8 semesters. In each course she has a probability 1/2 of getting an A. Assuming her grades in different courses are independent, what is the probability that she will have at least one semester with all A’s?
When Al and Bob play tennis, Al wins a set with probability 0.7 while Bob wins with probability 0.3.What is the probability that Al will be the first to win(a) Two sets. (b) Three sets?
Chevalier de M´er´e made money betting that he could “roll at least one 6 in 4 tries.” When people got tired of this wager he changed it to “roll at least one double 6 in 24 tries,” but then he started losing money. Compute the probabilities of winning these two bets.
Samuel Pepys wrote to Isaac Newton: “What is more likely,(a) At least one 6 in 6 rolls of one die.(b) At least two 6’s in 12 rolls?” Compute the probabilities of these events.
Suppose we roll two dice and let X and Y be the two numbers that appear. Find the distribution of |X − Y|.
Suppose we roll three tetrahedral dice that have 1, 2, 3, and 4 on their four sides. Find the distribution for the sum of the three numbers.
We roll two six-sided dice, one with sides 1, 2, 2, 3, 3, 4 and the other with sides 1, 3, 4, 5, 6, 8. What is the distribution of the sum?
How many children should a family plan to have so that the probability of having at least one child of each sex is ≥ 0.95?
How many times should a coin be tossed so that the probability of at least one head is ≥99%?
You want to invent a gambling game in which a person rolls two dice and is paid some money if the sum is 7, but otherwise he loses his money. How much should you pay him for winning a $1 bet if you want this to be a fair game, that is, to have expected value 0?
A bet is said to carry 3 to 1 odds if you win $3 for each $1 you bet. What must the probability of winning be for this to be a fair bet?
A lottery has one $100 prize, two $25 prizes, and five $10 prizes. What should you be willing to pay for a ticket if 100 tickets are sold?
In a popular gambling game, three dice are rolled. For a $1 bet you win $1 for each 6 that appears (plus your dollar back). If no 6 appears you lose your dollar. What is your expected value?
A roulette wheel has slots numbered 1 to 36 and two labeled with 0 and 00. Suppose that all 38 outcomes have equal probabilities. Compute the expected values of the following bets. In each case you bet one dollar and when you win you get your dollar back in addition to your winnings.(a) You win $1
In the Las Vegas game Wheel of Fortune, there are 54 possible outcomes.One is labeled “Joker,” one “Flag,” two “20,” four “10,” seven “5,” fifteen “2,” and twenty-four “1.” If you bet $1 on a number you win that amount of money if the number comes up (plus your dollar
Sic Bo is an ancient Chinese dice game played with three dice. One of the possibilities for betting in the game is to bet “big.” For this bet, you win if the total X is 11, 12, 13, 14, 15, 16, or 17, except when there are three 4’s or three 5’s. On a $1 bet on big, you win $1 plus your
Five people play a game of “odd man out” to determine who will pay for the pizza they ordered. Each flips a coin. If only one person gets heads (or tails) while the other four get tails (or heads) then he is the odd man and has to pay. Otherwise they flip again. What is the expected number of
A man and his wife decide that they will keep having children until they have one of each sex. Ignoring the possibility of twins and supposing that each trial is independent and results in a boy or a girl with probability 1/2, what is the expected value of the number of children they will have?
An unreliable clothes dryer dries your clothes and takes 20 minutes with probability 0.6 and buzzes for 4 minutes and does nothing with probability 0.4.If we assume that successive trials are independent and that we patiently keep putting our money in to try to get it to work, what is the expected
Suppose we pick a month at random from a non-leap year calendar and let X be the number of days in the month. Find the mean and variance of X.
The Elm Tree golf course in Cortland, NY, is a par 70 layout with 3 par fives, 5 par threes, and 10 par fours. Find the mean and variance of par on this course.
Can we have a random variable with E X = 3 and E X2 = 8?
Suppose P (X ∈ {1, 2, 3}) = 1 and E X = 2.5. What are the smallest and largest possible values for the variance?

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