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2021/8/17 HW1 – Intro to Probability

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Intro to Probability – Homework Assignment 1

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 How many plate numbers can be created in California format DLLLDDD, where D stands for digit and L stands for letter? What if no repetitions are allowed in the digits?

Problem 2 A zip code consists of 5 digits. If the first cannot be the digit 0, how many zip codes can be generated?

Problem 3 How many letter arrangements can be made from the letters of the word Iowa? How about from the letters of Connecticut?

Problem 4 How many 5-card poker hands are there? (A regular deck has 52 cards).

Problem 5 Assuming that you can only move to the right or down, in how many ways can you get from the top left point to the bottom right one along the depicted grid?

Problem 6 If you need to split a class of 32 students in 8 groups of 4, in how many ways can you do it?

2021/8/17 HW1 – Intro to Probability

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Problem 7 If a student is required to solve 8 out of 10 assigned problems, how many choices does she have? What if the first 4 problems are mandatory?

Problem 8 How many actual digit numbers exist? What if no two consecutive digits can be the same? How many numbers with at most digits exist? Numbers are to be understood as actuall numbers: 1 is a one digit number but 001 is not an actual number and certainly not a 3 digit number.

Problem 9 A total of 6 people need to be selected from a set of 10 twins of different families (20 people total). How many choices are there, if it is not allowed to select two siblings?

Problem 10 A total of 8 representatives need to be chosen from a group of 40 students, of which 24 are female and 16 are male. In how many ways can this be done? What if we require that there be at least 3 female and 3 male representatives?

n

n

2021/8/17 HW4 – Intro to Probability

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Intro to Probability – Homework Assignment 4

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is for ?

Problem 2 Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly 3 white balls?

Problem 3 Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second and third cards are spades.

Problem 4 In a certain community, 36% of the families own a dog and 22% of the families that own a dog also own a cat. In addition, 30% of the families own a cat. What is

a. the probability that a randomly selected family owns both a dog and a cat?

b. the conditional probability that a randomly selected family owns a dog given that it owns a cat?

Problem 5 Urn A contains 2 white and 4 red balls, whereas urn B contains 1 white and 1 red ball. A ball is randomly chosen from urn A and put into urn B, and a ball is then randomly selected from urn B. What is

a. the probability that the ball selected from urn B is white?

b. the conditional probability that the transferred ball was white given that a white ball is selected from urn B?

i i = 1, … , 12

2021/8/17 HW4 – Intro to Probability

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Problem 6 a. A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin?

b. Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin?

c. Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

Problem 7 Let . Express the following probabilities as simply as possible

Problem 8 A ball is in anyone of boxes and is in the -th box with probability . If the ball is in box , a search of that box will uncover it with probability . Show that the conditional probability that the ball is in box , given that a search of box did not uncover it, is

Problem 9 An event is said to carry negative information about an event iff

This is denoted by . Either give a proof or provide a counterexample for the following assertions

a. If , then .

b. If and , then .

c. If and , then .

Problem 10 The probability of getting a head on a single toss of a coin is . Suppose that A starts and continues to flip the coin until a tail shows up, at which point B starts flipping. Then B continues to flip until a tail comes up, at which point A takes over, and so on. Let

E ⊂ F

P (E|F ), P (E|F c), P (F |E), P (F |E c).

n i Pi i αi

j i

,  if j ≠ i,  and  ,  if i = j. Pj

1 − αiPi

(1 − αi)Pi 1 − αiPi

E

P (E|F ) ≤ P (E).

F ↘ E

F ↘ E E ↘ F

F ↘ E E ↘ G F ↘ G

F ↘ E G ↘ E F ∩ G ↘ G

p

Pn,m

2021/8/17 HW4 – Intro to Probability

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denote the probability that A accumulates a total of heads before B accumulates . Show that

n m

Pn,m = p Pn−1,m + (1 − p)(1 − Pm,n).

2021/8/17 HW2 – Intro to Probability

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Intro to Probability – Homework Assignment 2

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 An urn contains 3 balls: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 ball from the urn, replacing it in the box, and drawing a second ball. Describe the sample space of this experiment and for the experiment obtained not replacing the first ball in the urn.

Problem 2 In an experiment, a die is rolled continually until a 5 appears, at which point the experiment stops. What is the sample space of this experiment? Let denote the event that rolls are necessary to conclude the experiment. What outcomes are contained in

? What does the event represent?

Problem 3 Two dice are rolled. Using that is the event that the sum of the dice is odd, the event that at least one of the dice lands on 1, and the event that the sum is 5, describe the events , , , , and .

Problem 4 A hospital administrator codes incoming patients suffering from heart disease according to whether they have insurance (code 1 if they do and 0 if they do not) and based on their condition, which can be rated as good G, fair F, or serious S. Consider an experiment that consists of producing codes for such a patient.

a. Give the sample space of this experiment.

b. Let be the event that the patient is in serious condition and list all outcomes in .

c. Let be the event that the patient is uninsured and list all outcomes in .

d. List the outcomes in .

Problem 5

En n

En (⋃ ∞ n=1 En)

c

O O1 S5

O ∩ O1 O ∪ O1 O1 ∩ S5 O ∩ O c

1 O ∩ O1 ∩ S5

S S

U U

S c ∪ U

2021/8/17 HW2 – Intro to Probability

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Suppose that two events , are such that and satisfy and . What are the probabilities that

a. either or occurs?

b. occurs but does not?

c. both and occur?

What happens if you don’t assume that ?

Problem 6 At a summer camp three activities are offered: tennis, swimming, and archery. The courses are open to the 120 summer camp participants. There are 31, 24, and 17 in these activities, respectively. There are 4 participants in all activities, 11 in tennis and swimming, 9 in tennis and archery, 7 in swimming and archery.

a. If a participant is chosen at random, what is the probability that he or she is taking part exactly one activity?

b. If a participant is chosen at random, what is the probability that he or she is taking part in no activity?

c. If two participants are chosen at random, what is the probability that they are both taking part in exactly one activity?

Problem 7 Consider three fair dice with the following digits on their faces. The blue die has 2 faces each with 1, 5, and 9 on them. The red die shows the digits 3, 4, and 8 twice each and the green one 2,6, 7 twice each. Two players choose a die each and then roll it. The winner is the player who rolls the highest number. If you had the opportunity to choose whether to pick your die first, would you do it or rather go second?

Problem 8 Let E, F, and G be three events. Use them to describe the following events:

a. Of the three only occurs.

b. Both E and G occur , but F does not.

c. At least one event occurs of the three.

d. At least two of the events occur.

e. None of the events occur.

f. All occur.

E F E ∩ F = ∅ P (E) = 0.3 P (F ) = 0.5

E F

E F

E F

E ∩ F = ∅

F

2021/8/17 HW2 – Intro to Probability

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g. Exactly two events occur.

h. At most three of the events occur.

i. At most one of them occurs.

Problem 9 If and , show that . In general, prove Bonferroni’s inequality, i.e., that

Problem 10 An urn contains yellow and black balls. They are withdrawn one at a time until a total of yellow ones one are obtained. Find the probability that this happens on drawing number for any .

P (E) = 0.9 P (F ) = 0.8 P (E ∩ F ) ≥ 0.7

P (E ∩ F ) ≥ P (E) + P (F ) − 1

n m y ≤ n

k k ≤ m + n

2021/8/17 HW3 – Intro to Probability

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Intro to Probability – Homework Assignment 3

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 If 8 identical computers are to be divided among 4 lab rooms, how many divisions are possible? How many if each room must receive at least 1 computer?

Problem 2 Suppose that 10 fish are caught in a pond that contains 5 distinct types of fish.

a. How many different outcomes are possible, where an outcome consists in the number of fish caught of each of the 5 types?

b. How many outcomes are possible if 3 of the 10 fish caught are trout?

c. How many when at least 2 of the 10 are trout?

Problem 3 Prove that

by connecting the formula to a concrete situation (i.e. interpreting its meaning). Show that the formula implies

Problem 4 Consider a tournament of contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players that tied for first place, the next group being those that tied for the next-best position, and so on. Let denote the number of different possible outcomes. For instance, , since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they

( ) = ( )( ) + ( )( ) + ⋯ + ( )( ) n + m

r

n

0

m

r

n

1

m

r − 1

n

r

m

0

( ) = ( ) 2

+ ( ) 2

+ ⋯ + ( ) 2

. 2n

n

n

0

n

1

n

n

n

N (n) N (2) = 3

2021/8/17 HW3 – Intro to Probability

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could tie for first.

a. List all possible outcomes when .

b. Setting , show that

without resorting to any computation.

c. Find for .

Problem 5 In how many ways can identical balls be placed into urns so that the th urn contains at least balls, for each ? Assume that .

Problem 6 A, B, and C take turns flipping a coin in that order.The first one to get a head wins. Explain why the sample space of this experiment can be defined by

by describing what it means. Then use it to describe the following events:

Problem 7 A retail establishment accepts either the American Express or the VISA credit card. A total of 24% of its customers carry an American Express card, 61% carry a VISA card, and 11% carry both cards. What percentage of its customers carry a credit card that the establishment will accept?

Problem 8 Two cards are randomly selected from an ordinary playing deck (52 cards). What is the probability that one of the cards is an ace and the other one is either a ten, a jack, a queen, or a king? This combinations is called a black jack.

Problem 9 Prove that

n = 3

N (0) = 1

N (n) = n

∑ i=1

( )N (n − i) = n−1

∑ i=0

( )N (i), n

i

n

i

N (n) n = 3, 4

n r i

mi i = 1, … , r n ≥ ∑ r

i=1 mi

1, 01, 001, 0001, 00001, …

WA = ‘‘A wins “, WB = ‘‘B wins “,  and (WA ∪ WB) c.

P (E ∩ F c ) = P (E) − P (E ∩ F ).