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A pizza shop offers three toppings: pineapple, peppers, and pepperoni. A pizza can have 0, 1, 2, or 3 toppings. Consider the probability that a random customer asks for two toppings.

(i) The random experiment 

(ii) Sample space 

(iii) Event

(iv) Random variable. 

Express the probability in question in terms of the defined random variable, but do not compute the probability.

Roll four dice. Consider the probability of getting all fives.

(i) The random experiment, 

(ii) Sample space, 

(iii) Event, and 

(iv) Random variable. 

Express the probability in question in terms of the defined random variable, but do not compute the probability.

In Angel’s garden, there is a 3% chance that a tomato will be bad. Angel harvests 100 tomatoes and wants to know the probability that at most five tomatoes are bad.

(i) The random experiment.

(ii) Sample space. 

(iii) Event.

(iv) Random variable. 

Express the probability in question in terms of the defined random variable, but do not compute the probability.

Your friend was sick and unable to make today’s class. Explain to your friend, using your own words, the meaning of the terms: 

(i) Random experiment, 

(ii) Sample space, 

(iii) Event, and 

(iv) Random variable.

Bored one day, you decide to play the video game Angry Birds until you win.Every time you lose, you start over. Consider the probability that you win in less than 1000 tries.

(i) The random experiment 

(ii) Sample space 

(iii) Event

(iv) Random variable 

Express the probability in question in terms of the defined random variable, but do not compute the probability.

In two dice rolls, let X be the outcome of the first die, and Y the outcome of the second die. Then X + Y is the sum of the two dice. Describe the following events in terms of simple outcomes of the random experiment:
(a) {X + Y = 4}.
(b) {X + Y = 9}.
(c) {Y = 3}.
(d) {X = Y}.
(e) {X > 2Y}.

A bag contains r red and b blue balls. You reach into the bag and take k balls. Let R be the number of red balls you take. Let B be the number of blue balls. Express the following events in terms of the random variables R and B:
(a) You pick no red balls.
(b) You pick one red and two blue balls.
(c) You pick four balls.
(d) You pick twice as many red balls as blue balls.

A couple plans to continue having children until they have a girl or until they have six children, whichever comes first. Describe a sample space and a reasonable random variable for this random experiment.

A sample space has four elements ω1, . . . , ω4 such that ω1 is twice as likely as ω2, which is three times as likely as ω3, which is four times as likely as ω4. Find the probability function.

A random experiment has three possible outcomes a, b, and c, with P(a) = p, P(b) = p2, and P(c) = p. What choice(s) of p makes this a valid probability model?

Let P be a probability function on Ω = {a, b} such that P(a) = p and P(b) = 1 − p for 0 ≤ p ≤ 1. Let Q be a function on Ω defined by Q(ω) = [P(ω)]2. For what value(s) of p will Q be a valid probability function?

A club has 10 members and is choosing a president, vice-president, and treasurer. All selections are equally likely.
(a) What is the probability that Tom is selected president?
(b) What is the probability that Brenda is chosen president and Liz is chosen treasurer?

A fair coin is flipped six times. What is the probability that the first two flips are heads and the last two flips are tails? Use the multiplication principle.

Let P1 and P2 be two probability functions on Ω. Define a new function P such that P(A) = (P1(A)+P2(A))/2. Show the P is a probability function.

Suppose that license plates can be two, three, four, or five letters long, taken from the alphabets A to Z. All letters are possible, including repeats. A license plate is chosen at random in such a way so that all plates are equally likely.
(a) What is the probability that the plate is “A-R-R?”
(b) What is the probability that the plate is four letters long?
(c) What is the probability that the plate is a palindrome?
(d) What is the probability that the plate has at least one “R?”

Suppose you throw five dice and all outcomes are equally likely.
(a) What is the probability that all dice are the same?
(b) What is the probability of getting at least one 4?
(c) What is the probability that all the dice are different?

Amy is picking her fall term classes. She needs to fill three time slots, and there are 20 distinct courses to choose from, including probability 101, 102, and 103. She will pick her classes at random so that all outcomes are equally likely.
(a) What is the probability that she will get probability 101?
(b) What is the probability that she will get probability 101 and Probability 102?
(c) What is the probability she will get all three probability courses?

Suppose k numbers are chosen from {1, . . . , n}, where k < n, sampling without replacement. All outcomes are equally likely. What is the probability that the numbers chosen are in increasing order?

Suppose P(A) = 0.40, P(B) = 0.60, and P(A or B) = 0.80. Find
(a) P(neither A nor B occur).
(b) P(AB).
(c) P(one of the two events occurs, and the other does not).

Suppose A and B are mutually exclusive, with P(A) = 0.30 and P(B) = 0.60. Find the probability that
(a) At least one of the two events occurs
(b) Both of the events occur
(c) Neither event occurs
(d) Exactly one of the two events occur

Suppose P(A ∪ B) = 0.6 and P(A ∪ Bc) = 0.8. Find P(A).

Suppose X is a random variable that takes values on all positive integers. Let  A = {2 ≤ X ≤ 4} and B = {X ≥ 4}. Describe the events  

(i) Ac

(ii) Bc

(iii) AB

(iv) A ∪ B

Suppose X is a random variable that takes values on {0, 0.01, 0.02, . . . , 0.99, 1}. If each outcome is equally likely, find
(a) P(X ≤ 0.33).
(b) P(0.55 ≤ X ≤ .66).

Let A, B, C, be three events. At least one event always occurs. But it never happens that exactly one event occurs. Nor does it ever happen that all three events occur. If P(AB) = 0.10 and P(AC) = 0.20, find P(B).

See the assignment of probabilities to the Venn diagram in Figure 1.4. Find the following:

(a) P(No events occur).
(b) P(Exactly one event occurs).
(c) P(Exactly two events occur).
(d) P(Exactly three events occur).
(e) P(At least one event occurs).
(f) P(At least two events occur).
(g) P(At most one event occurs).
(h) P(At most two events occur).

Four coins are tossed. Let A be the event that the first two coins comes up heads. Let B be the event that the number of heads is odd. Assume that all 16 elements of the sample space are equally likely. Describe and find the probabilities of 

(i) AB,

(ii) A ∪ B,

(iii) ABc.

Two dice are rolled. Let X be the maximum number obtained. (Thus, if 1 and 2 are rolled, X = 2; if 5 and 5 are rolled, X = 5.) Assume that all 36 elements of the sample space are equally likely. Find the probability function for X. That is, find P(X = x), for x = 1, 2, 3, 4, 5, 6.

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 number rolled is the number on the base. In five rolls, what is the probability of rolling at least one 2?

Let

(a) Show that Q is a probability function. That is, show that the terms are non-negative and sum to 1.
(b) Let X be a random variable such that P(X = k) = Q(k), for k = 0, 1, 2, . . . . Find P(X > 2) without summing an infinite series.

The function
is a probability function for some choice of c. Find c.

Let A, B, C be three events. Find expressions for the events:
(a) At least one of the events occurs.
(b) Only B occurs.
(c) At most one of the events occurs.
(d) All of the events occur.
(e) None of the events occur.

The odds in favor of an event is the ratio of the probability that the event occurs to the probability that it will not occur. For example, the odds that you were born on a Friday, assuming birth days are equally likely, is 1 to 6, often written 16 or 1 to 6.
(a) In Texas Hold’em Poker, the odds of being dealt a pair (two cards of the same denomination) is 116. What is the chance of not being dealt a pair?
(b) For sporting events, bookies usually quote odds as odds against, as opposed to odds in favor. In the Kentucky Derby horse race, our horse Daddy Long Legs was given 2–9 odds. What is the chance that Daddy Long Legs wins the race?

An exam had three questions. One-fifth of the students answered the first question correctly; one-fourth answered the second question correctly; and one-third answered the third question correctly. For each pair of questions, one-tenth of the students got that pair correct. No one got all three questions right. Find the probability that a randomly chosen student did not get any of the questions correct.

Suppose P(ABC) = 0.05, P(AB) = 0.15, P(AC) = 0.2, P(BC) = 0.25, P(A) = P(B) = P(C) = 0.5. For each of the events given next, write the event using set notation in terms of A, B, and C, and compute the corresponding probability.
(a) At least one of the three events A,B,C occur.
(b) At most one of the three events occurs.
(c) All of the three events occurs.
(d) None of the three events occurs.
(e) At least two of the three events occurs.
(f) At most two of the three events occurs.

Find the probability that a random integer between 1 and 5000 is divisible by 4, 7 or 10.

(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 six squares of a two-by-three checkerboard is randomly colored red or black. Find the probability that at least one of the three columns of the checkerboard is all red.

Given events A and B, show that the probability that exactly one of the events occurs equals 

P(A) + P(B) − 2P(AB).

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).

Modify the code in the R script CoinFlip.R to simulate the probability of getting exactly one head in four coin tosses.

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.

Use R to simulate the probability of getting at least one 8 in the sum of two dice rolls.

Use R to simulate the probability in Exercise 1.30


Data 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 number rolled is the number on the base. In five rolls, what is the probability of rolling at least one 2?

Suppose P(A) = P(B) = 0.3 and P(A|B) = 0.5. Find P(A ∪ B).

Suppose P(A) = P(B) = pand P(A U B) = p2. Find P(A|B).

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 probability of getting all heads?

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.

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 8’s, as in Figure 2.8. 

Suppose dice A and B are rolled. Show that A is more likely to get the higher number. That is, P(A > B) > 0.50, where {A > B} denotes the event that A beats B. 

Now show that if B and C are rolled, B is more likely to get the higher number. And, remarkably, if C and A are rolled, C is more likely to get the higher number. 

Many relationships in life are transitive. For instance, if Amy is taller than Ben and Ben is taller than Charlie, then Amy is taller than Charlie. But these dice show that the relation “more likely to roll a higher number” is not transitive. 

The dice are the basis of a magic trick. You pick any die. Then I can always pick a die that is more likely to beat yours. If you pick A, I pick C. If you pick B, I pick A. And if you pick C, I pick B.

(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 exhibit a counter-example.

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?

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.

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 (adapted from Blom et al., 1991).
(a) Obtain an expression for p1, the probability that he gets both questions right conditional on getting the first question right.
(b) Obtain an expression for p2, the probability that he gets both questions right conditional on getting either of the two questions right (A or B).
(c) Show that p2 ≤ p1. This may seem paradoxical. Knowledge that A or B has taken place makes the conditional probability that A and B happens smaller than when we know that A has happened. Can you untangle the paradox?

Suppose P(A) = 1/2, P(Bc|AC) = 1/3 and P(C|A) = 1/4. Find P(ABC).

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).

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 becomes at least 50%.

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?

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 to symmetry. That is, make a qualitative argument for why the answer should not depend on the heart suit.

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 uniformly at random and then picks a pack of candy. What is the probability that the pack chosen is Gummi Bears? Solve 

(i) By using a tree diagram and 

(ii) By another method.

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 that the Bi’s partition the sample space. Illustrate the law of total probability by writing out formula 2.9 and finding the probabilities for each term in the equation.

Give a formula for P(A|Bc) in terms of P(A), P(B), and P(AB) only.

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 collection of 72 challenging, and sometimes amusing, mathematical problems, several of which involve probability. Here is Problem #5. A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?

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 and describe the events {X = k} for k = 1, 5, 10, 25.
(b) Assume that coin selections are equally likely. Find the probabilities for each of the aforementioned four events.
(c) What is the probability that Judith’s coin is worth more than Joe’s? (It is not 1/2.)

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 creates a frequency table showing how many people have each birthday, and then determines if two is in that table; that is, whether two or more people have the same birthday. Use and suitably modify the expression for the following problems.
(a) Simulate the probability that two people have the same birthday in a room of 23 people.
(b) Estimate the number of people needed so that the probability of a match is 95%.
(c) Find the approximate probability that three people have the same birthday in a room of 50 people.
(d) Estimate the number of people needed so that the probability that three people have the same birthday is 50%.

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 probability that the chosen die is the fair one.

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% reliable. That is, the witness will correctly identify the color of a cab 80% of the time. What is the probability that the cab actually was blue?

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 a person telling a lie will fail the polygraph test.
(a) Suppose a population consists of 5% liars. A random person takes a polygraph test, which concludes that they are lying. What is the probability that they are actually lying?
(b) Consider the probability that a person is actually lying given that the polygraph says that they are. Using the definition of reliability, how reliable must the polygraph test be in order that this probability is at least 80%?

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 over 50 and a specificity of about 95%. That is, the false-negative rate is 15% and the false-positive rate is 5%. If a woman over 50 has a mammogram and it comes back positive for breast cancer, what is the probability that she has the disease?

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 being a smoker is 35%. Find the probability of being a smoker if neither parent is a smoker.

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.

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 B. Draw a tree diagram for this problem and use it to find the probability that the final ball picked is white.

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). Find the probability that X is odd. (Consider Taylor expansions of eλ and e−λ.)

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, Blaise Pascal and Pierre de Fermat, considered the probability that in 24 throws of a pair of dice at least one “double six” occurs. It was commonly believed by gamblers at the time that betting on double sixes in 24 throws would be a profitable bet (i.e., greater than 50% chance of occurring). But Pascal and Fermat showed otherwise. Find this probability.

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.

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 each other. Show that the total number of eggs that develop has a Poisson distribution with parameter λp. Condition on the number of eggs that are laid.

Give a probabilistic interpretation of the series

That is, pose a probability question for which the sum of the series is the answer.

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 left-top corner of the chessboard is hit by exactly two darts.
(b) Find an approximation of this probability using an appropriate distribution.

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 probability that X is between one and three using

(a) The exact distribution of X.
(b) An approximate distribution of X.

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 = 4. Each pill is equally likely. Suppose you get three colds next winter. What is the probability you took the blue pill?

Suppose X ∼ Pois(λ). Find the probability that X is odd. (Consider Taylor expansions of eλ and e−λ.)

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 no-hitter baseball games that were pitched in the 104 baseball seasons between 1901 and 2004 are given in Huber and Glen (2007). For instance, 18 seasons saw no no-hit games pitched; 30 seasons saw one no-hit game, etc. Use these data to model the number of no-hit games for a baseball season. Create a table that compares the observed counts with the expected number of no-hit games under your model.

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?)

Suppose A and B are independent events. Show that Ac and Bc are independent events.

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 it has bark disease. Find the probability that a tree has root rot.

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) = P(A)P(B)P(C), but no pair of events is independent.

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 throws of a pair of dice at least one “double six” occurs. It was commonly believed by gamblers at the time that betting on double sixes in 24 throws would be a profitable bet (i.e., greater than 50% chance of occurring). But Pascal and Fermat showed otherwise. Find this probability.

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 winning to be at least 50%?

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 the number of cherries that show up from one play of the slot machine. Find P(X = k), for k = 0, 1, 2, 3. Slot machines are also known as “one-armed bandits.”

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 disease, what is the probability the queen is a carrier.
(b) If there is a fourth prince, what is the probability that he will have hemophilia?

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, . . . , 8.

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 people needed so that the probability of at least one match is 95%. Show k ≈ 2.45√c.

(b) Suppose there are m categories, all of which are independent and take c possible values. Let k be the number of people needed so that the probability of at least one match in any category is 95%. Show k ≈ 2.45 √c/m.

(c) A group of k people is comparing 

(i) Their birthdays, 

(ii) The last two digits on their social security card, and 

(iii) The two-digit ticket number on their movie stubs. 

How big should k be so that there is a 50% chance of at least one match? A 95% chance?

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.

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, what is the chance that one shoe from each pair will be represented? (Remember, a left shoe is different than a right shoe.)

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 three).
(a) Show that the suit distribution 4-4-3-2 is more likely than 4-3-3-3.
(b) In fact, besides the 4-4-3-2 distribution, there are three other patterns of suit distributions that are more likely than 4-3-3-3. Can you find them?

Find the probability that a bridge hand contains a nine-card suit. That is, the number of cards of the longest suit is nine.

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 that there will be exactly one piece in each row and in each column?

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 the game, but the ignorance of the people is fearful. Why, I have known clergymen, good men, kind-hearted, liberal, sincere, and all that, who did not know the meaning of a “flush.” It is enough to make one ashamed of one’s species.
Find the probabilities for the following poker hands. They are arranged in decreasing order of probability.

(a) Straight flush. (Five cards in a sequence and of the same suit.)

(b) Four of a kind. (Four cards of one face value and one other card.)

(c) Full house. (Three cards of one face value and two of another face value.)

(d) Flush. (Five cards of the same suit. Does not include a straight flush.)

(e) Straight. (Five cards in a sequence. Does not include a straight flush. Ace can be high or low.)

(f) Three of a kind. (Three cards of one face value. Does not include four of a kind or full house.)

(g) Two pair. (Does not include four of a kind or full house.)

(h) One pair. (Does not include any of the aforementioned conditions.)

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) Suppose a walker starts at the origin (0, 0) and at each discrete unit of time moves either up one unit or to the right one unit each with probability 1/2. If x > y, find the probability that a walk from (0,0) to (x,y) always stays above the main diagonal.

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 the following probabilities and solve them with R.
(a) The probability that exactly 10 people are infected.
(b) That probability that at least 16 people are infected.
(c) The probability that between 12 and 14 people are infected.
(d) The probability that someone is infected.

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 least one 6 appears.
2. Twelve fair dice are tossed and at least two 6’s appear.
3. Eighteen fair dice are tossed and at least three 6’s appear.

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 choose pizza. Let X be the number of times she chose pizza last week.
(b) Brenda plays basketball, and there is a 60% she makes a free throw. Let X be the number of successful baskets she makes in a game.
(c) A bowl contains 100 red candies and 150 blue candies. Carl reaches and takes out a sample of 10 candies. Let X be the number of red candies in his sample.
(d) Evan is reading a 600-page book. On even-numbered pages, there is a 1% chance of a typo. On odd-numbered pages, there is a 2% chance of a typo. Let X be the number of typos in the book.
(e) Fran is reading a 600-page book. The number of typos on each page has a Bernoulli distribution with p = 0.01. Let X be the number of typos in the book.

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 isolated if its degree is 0. Find the probability that a particular vertex is isolated.

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