Questions and Answers of Probability With Applications

A fair coin is tossed until heads appears four times.(a) Find the probability that it took exactly 10 flips.(b) Find the probability that it took at least 10 flips.(c) Let Y be the number of tails
Baseball teams A and B face each other in the World Series. For each game, the probability that A wins the game is p, independent of other games. Find the expected length of the series.
Let X and Y be independent geometric random variables with parameter p. Find the pmf of X + Y.
(i) In a bridge hand of 13 cards, what is the probability of being dealt exactly two red cards? (ii) In a bridge hand of 13 cards, what is the probability of being dealt four spades, four clubs,
People whose blood type is O-negative are universal donors—anyone can receive a blood transfusion of O-negative blood. In the U.S., 7.2% of the people have O-negative blood. A blood donor clinic
Andy and Beth are playing a game worth $100. They take turns flipping a penny. The first person to get 10 heads will win. But they just realized that they have to be in math class right away and are
Let R ∼ Geom(p). Conditional on R = r, suppose X has a negative binomial distribution with parameters r and p. Show that the marginal distribution of X is geometric. What is the parameter?
There are 500 deer in a wildlife preserve. A sample of 50 deer are caught and tagged and returned to the population. Suppose that 20 deer are caught.(a) Find the mean and standard deviation of the
The Lady Tasting Tea is one of the most famous experiments in the founding history of statistics. In his 1935 book The Design of Experiments (1935), Sir Ronald A. Fisher writes, A Lady declares that
In a town of 20,000, there are 12,000 voters, of whom 6000 are registered democrats and 5000 are registered republicans. An exit poll is taken of 200 voters. Assume all registered voters actually
Consider the hypergeometric distribution with parameters r, n, and N. Suppose r depends on N in such as way that r/N → p as N → ∞, where 0 < p < 1. Show that the mean and variance of the
A Halloween bag contains three red, four green, and five blue candies. Tom reaches in the bag and takes out three candies. Let R, G, and B denote the number of red, green, and blue candies,
In a city of 100,000 voters, 40% are Democrat, 30% Republican, 20% Green, and 10% Undecided. A sample of 1000 people is selected.(a) What is the expectation and variance for the number of Greens in
A random experiment takes r possible values, with respective probabilities p1, . . . ,pr. Suppose the experiment is repeated N times, where N has a Poisson distribution with parameter λ. For k = 1,
Suppose a gene allele takes two forms A and a, with P(A) = 0.20 = 1 − P(a). Assume a population is in Hardy–Weinberg equilibrium.(a) Find the probability that in a sample of eight
In 10 rolls of a fair die, let X be the number of fives rolled, let Y be the number of even numbers rolled, and let Z be the number of odd numbers rolled.(a) Find Cov(X, Y).(b) Find Cov(X, Z).
A professor starts each class by picking a number from a hat that contains the numbers 1–30. If a prime number is chosen, there is no homework that day. There are 42 class periods in the semester.
Prove the identitywhere the sum is over all nonnegative integers that sum to 4. п! Σ 4" ala2!аз!а,!" ai+a4=n
Find the expectation and variance of the Benford’s law distribution.
An elevator containing p passengers is at the ground floor of a building with n floors. On its way to the top of the building, the elevator will stop if a passenger needs to get off. Passengers get
Give a combinatorial argument (not an algebraic one) for why ()-(E)
A joint probability mass function is given byP(X = 1, Y = 1) = 1/8 P(X = 1, Y = 2) = 1/4.P(X = 2, Y = 1) = 1/8 P(X = 2, Y = 2) = 1/2.(a) Find the marginal distributions of X and Y.(b) Are X and Y
Suppose P1, . . . , Pk are probability functions on Ω. Let a1, . . . , ak be a sequence of numbers. Under what conditions on the ais will P = a1P1 + · · · + akPk be a probability
The following problem appeared in the news column “Ask Marilyn” on September 19, 2010. Four identical sealed envelopes are on a table. One of them contains a $100 bill. You select an
Simulate the nontransitive dice probabilities in Exercise 2.6. Data from exercise 2.6: Consider three nonstandard dice. Instead of the numbers 1 through 6, die A has two 3??s, two 5??s, and two
Suppose A, B, and C are independent events with respective probabilities 1/3, 1/4, and 1/5. Find(a) P(ABC)(b) P(A or B or C)(c) P(AB|C)(d) P(B|AC)(e) P(At most one of the three events occurs).
A bag of 16 balls contains one red, three yellow, five green, and seven blue balls. Suppose four balls are picked, sampling with replacement.(a) Find the probability that the sample contains at least
Hemocytometer slides are used to count cells and other microscopic particles. They consist of glass engraved with a laser-etched grid. The size of the grid is known making it possible to count the
The number of eggs a chicken hatches is a Poisson random variable. The probability that the chicken hatches no eggs is 0.10. What is the probability that she hatches at least two eggs?
Computer scientists have modeled the length of search queries on the web using a Poisson distribution. Suppose the average search query contains about three words. Let X be the number of words in a
Suppose X has a Poisson distribution and P(X = 2) = 2P(X = 1). Find P(X = 3).
I have two dice, one is a standard die. The other has three ones and three fours. I flip a coin. If heads, I will roll the standard die five times. If tails, I will roll the other die five times. Let
A new experimental drug is given to patients suffering from severe migraine headaches. Patients report their pain experience on a scale of 1 – 10 before and after the drug. The difference of their
Ecologists use occupancy models to study animal populations. Ecologists at the Department of Natural Resources use helicopter surveying methods to look for otter tracks in the snow along the
See Example 3.26. Consider a random graph on n = 8 vertices with edge probability p = 0.25.(a) Find the probability that the graph has at least six edges.(b) A vertex of a graph is said to be
For the following situations, identify whether or not X has a binomial distribution. If it does, give n and p; if not, explain why.(a) Every day Amy goes out for lunch there is a 25% chance she will
In 1693, Samuel Pepys wrote a letter to Isaac Newton posing the following question. Which of the following three occurrences has the greatest chance of success?1. Six fair dice are tossed and at
Every person in a group of 1000 people has a 1% chance of being infected by a virus. The process of being infected is independent from person to person.Using random variables, write expressions for
A walk in the positive quadrant of the plane consists of a sequence of moves, each one from a point (a, b) to either (a + 1, b) or (a, b + 1).(a) Show that the number of walks from the origin:(b)
There are few things that are so unpardonably neglected in our country as poker. The upper class knows very little about it. Now and then you find ambassadors who have sort of a general knowledge of
A chessboard is an eight-by-eight arrangement of 64 squares. Suppose eight chess pieces are placed on a chessboard at random so that each square can receive at most one piece. What is the probability
Find the probability that a bridge hand contains a nine-card suit. That is, the number of cards of the longest suit is nine.
Many bridge players believe that the most likely distribution of the four suits (spades, hearts, diamonds, and clubs) in a bridge hand is 4-3-3-3 (four cards in one suit, and three cards of the other
There are 40 pairs of shoes in Bill’s closet. They are all mixed up.(a) If 20 shoes are picked, what is the chance that Bill’s favorite sneakers will be in the group?(b) If 20 shoes are picked,
Suppose X1,X2,X3 are i.i.d. random variables, each uniformly distributed on {1, 2, 3}. Find the probability function for X1 + X2 + X3. That is, find P(X1 + X2 + X3 = k), for k = 3, . . . , 9.
Concidences (Diaconis and Mosteller, 1989). See Section 2.3.1 on the birthday problem. Some categories (like birthdays) are equally likely to occur, with c possible values.(a) Let k be the number of
Let X be a random variable such that P(X = k) = k/10, for k = 1, 2, 3, 4. Let Y be a random variable with the same distribution as X. Suppose X and Y are independent. Find P(X + Y = k), for k = 2, .
There is a 50-50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia.(a) If the queen has had three princes without
The original slot machine had 3 reels with 10 symbols on each reel. On each play of the slot machine, the reels spin and stop at a random position. Suppose each reel has one cherry on it. Let X be
A lottery will be held. From 1000 numbers, one will be chosen as the winner. A lottery ticket is a number between 1 and 1000. How many tickets do you need to buy in order for the probability of
A gambler’s dispute in 1654 is said to have led to the creation of mathematical probability. Two French mathematicians, Blaise Pascal and Pierre de Fermat, considered the probability that in 24
Toss two dice. Let A be the event that the first die rolls 1, 2, or 3. Let B be the event that the first die rolls 3, 4, or 5. Let C be the event that the sum of the dice is 9. Show that P(ABC) =
There is a 70% chance that a tree is infected with either root rot or bark disease. The chance that it does not have bark disease is 0.4. Whether or not a tree has root rot is independent of whether
Suppose A and B are independent events. Show that Ac and Bc are independent events.
Cars pass a busy intersection at a rate of approximately 16 cars per minute. What is the probability that at least 1000 cars will cross the intersection in the next hour? (What is the rate per hour?)
Table 3.10 from Huber and Gleu (2007) shows the number of no hitter baseball games that were pitched in the 104 ball seasons between 1901 and 2004. For instance, the following data on the number of
Suppose X ∼ Pois(λ). Find the probability that X is odd. (Consider Taylor expansions of eλ and e−λ.)
If you take the red pill, the number of colds you get next winter will have a Poisson distribution with λ = 1. If you take the blue pill, the number of colds will have a Poisson distribution with λ
A physicist estimated that the probability of a U.S. nickel landing on its edge is one in 6000. Suppose a nickel is flipped 10,000 times. Let X be the number of times it lands on its edge. Find the
A chessboard is put on the wall and used as a dart board. Suppose 100 darts are thrown at the board and each of the 64 squares is equally likely to be hit.(a) Find the exact probability that the
Give a probabilistic interpretation of the seriesThat is, pose a probability question for which the sum of the series is the answer. 4!e +... 2!e 6!e
Suppose that the number of eggs that an insect lays is a Poisson random variable with parameter λ. Further, the probability that an egg hatches and develops is p. Egg hatchings are independent of
Which is more likely: 5 heads in 10 coin flips, 50 heads in 100 coin flips, or 500 heads in 1000 coin flips? Use R’s dbinom command to find out.
Simulate the 1654 gambler’s dispute in Exercise 3.5.Data from Exercise 3.5.A gambler’s dispute in 1654 is said to have led to the creation of mathematical probability. Two French mathematicians,
Choose your favorite value of λ and let X ∼ Pois(λ). Simulate the probability that X is odd. See Exercise 3.35. Compare with the exact solution.Data from Exercise 3.35.Suppose X ∼ Pois(λ).
Box A contains one white ball and two red balls. Box B contains one white ball and three red balls. A ball is picked at random from box A and put into box B. A ball is then picked at random from box
Consider flipping coins until either two heads HH or heads then tails HT first occurs. By conditioning on the first coin toss, find the probability that HT occurs before HH.
In a certain population of youth, the probability of being a smoker is 20%. The probability that at least one parent is a smoker is 30%. And if at least one parent is a smoker, the probability of
According to the National Cancer Institute, for women between 50 and 59, there is a 2.38% chance of being diagnosed with breast cancer. Screening mammography has a sensitivity of about 85% for women
A polygraph (lie detector) is said to be 90% reliable in the following sense: There is a 90% chance that a person who is telling the truth will pass the polygraph test, and there is a 90% chance that
An eyewitness observes a hit-and-run accident in New York City, where 95% of the cabs are yellow and 5% are blue. A witness asserts the cab was blue. A police expert believes the witness is 80%
Your friend has three dice. One die is fair. One die has fives on all six sides. One die has fives on three sides and fours on three sides. A die is chosen at random. It comes up five. Find the
The R command> sample(1:365,23,replace=T)simulates birthdays from a group of 23 people. The expression> 2 %in% table(sample(1:365,23,replace=T))can be used to simulate the birthday problem. It
Judith has a penny, nickel, dime, and quarter in her pocket. So does Joe. They both reach into their pockets and choose a coin. Let X be the greater (in cents) of the two.(a) Construct a sample space
Lewis Carroll, author of Alice’s Adventures in Wonderland, is the pen name of Charles Lutwidge Dodgson, who was an Oxford mathematician and logician. Lewis Carroll’s Pillow Problems (1958), is a
Give a formula for P(A|Bc) in terms of P(A), P(B), and P(AB) only.
In a roll of two tetrahedron dice, each labeled one to four, let X be the sum of the dice. Let A = {X is prime} and B1 = {X = 2}, B2 = {3 ≤ X ≤ 5}, B3 = {6 ≤ X ≤ 7}, and B4 = {X = 8}. Observe
Amy has two bags of candy. The first bag contains two packs of M&Ms and three packs of Gummi Bears. The second bag contains four packs of M&Ms and two packs of Gummi Bears. Amy chooses a bag
A standard deck of cards has one card missing. A card is then picked from the deck. What is the chance that it is a heart? Solve this problem in two ways:(a) Condition on the missing card.(b) Appeal
Jimi has 5000 songs on his iPod shuffle, which picks songs uniformly at random. Jimi plans to listen to 100 songs today. What is the chance he will hear at least one song more than once?
The planet Mars revolves around the sun in 687 days. Answer Von Mises’ birthday question for Martians. That is, how many Martians must be in a room before the probability that some share a birthday
Prove the addition rule for conditional probabilities. That is, show that for events A, B, and C, P(A U B|C) = P(A|C) + P(B|C) − P(AB|C).
Suppose P(A) = 1/2, P(Bc|AC) = 1/3 and P(C|A) = 1/4. Find P(ABC).
Bob is taking a test. There are two questions he is stumped on and he decides to guess. Let A be the event that he gets the first question right; let B be the event he gets the second question right
In the game of Poker, a flush is five cards of the same suit. Use conditional probability to find the probability of being dealt a flush.
A bag of 15 Scrabble tiles contains three each of the letters A, C, E, H, and N. If you pick six letters one at a time, what is the chance that you spell C-H-A-N-C-E?
(a) True or false: P(A|B) + P(A|Bc) = 1. Either show it true for any event A and B or exhibit a counter-example.(b) True or false: P(A|B) + P(Ac|B) = 1. Either show it true for any event A and B or
Consider three nonstandard dice. Instead of the numbers 1 through 6, die A has two 3??s, two 5??s, and two 7??s; die B has two 2??s, two 4??s, and two 9??s; and die C has two 1??s, two 6??s, and two
Find a simple expression for P(A|B) under the following conditions:(a) A and B are disjoint.(b) A = B.(c) A implies B.(d) B implies A.
John flips three pennies.(a) Amy peeks and sees that the first coin lands heads. What is the probability of getting all heads?(b) Zach peeks and sees that one of the coins lands heads. What is the
Suppose P(A) = P(B) = p1 and P(A U B) = p2. Find P(A|B).
Suppose P(A) = P(B) = 0.3 and P(A|B) = 0.5. Find P(A ∪ B).
Use R to simulate the probability in Exercise 1.30Data from Exercise 1.30:A tetrahedron dice is four-sided and labeled with 1, 2, 3, and 4. When rolled it lands on the base of a pyramid and the
Use R to simulate the probability of getting at least one 8 in the sum of two dice rolls.
Modify the code in the R script Divisible356.R to simulate the probability that a random integer between 1 and 5000 is divisible by 4, 7, or 10.
Modify the code in the R script CoinFlip.R to simulate the probability of getting exactly one head in four coin tosses.
Given events A, B, C, show that the probability that exactly one of the events occurs equals: P(A) + P(B) + P(C) − P(AB) − P(AC) − P(BC) + 3P(ABC).
Given events A and B, show that the probability that exactly one of the events occurs equals P(A) + P(B) − 2P(AB).
(a) Each of the four squares of a two-by-two checkerboard is randomly colored red or black. Find the probability that at least one of the two columns of the checkerboard is all red.(b) Each of the
Find the probability that a random integer between 1 and 5000 is divisible by 4, 7 or 10.

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