Axioms of Probability

Two balls are chosen randomly from an urn containing$8$white,$4$ black, and$2$ orange balls. Suppose that we win $\backslash (2$for each black ball selected and we lose $\backslash )1$ for each white ball selected. Let $X$ denote our winnings. What are the possible values of $X$, and what are the probabilities associated with each value?

A box contains $3$ marbles: $1$ red, $1$ green, and $1$ blue. Consider an experiment that consists of taking $1$ marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

In an experiment, die is rolled continually until a $6$ appears, at which point the experiment stops. What is the sample space of this experiment? Let $E_n$ denote the event that $n$ rolls are necessary to complete the experiment. What points of the sample space are contained in $E_n$? What is ${\left(\underset{1}{\overset{\infty }{{\cup E}_n}}\right)}^c$?

Two dice are thrown. Let $E$ be the event that the sum of the dice is odd, let $F$ be the event that at least one of the dice lands on $1$, and let $G$ be the event that the sum is $5$. Describe the events $EF, E\cup F,FG, E{F}^c, and EFG$.

A, B, and C take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined by

$S = \left\{\begin{array}{l}1, 01, 001, 0001, . . . , \\ 0000 · · ·\end{array}\right.$

(a) Interpret the sample space.

(b) Define the following events in terms of S:

(i) $A$ wins = $A$.

(ii) $B$ wins = $B$.

(iii) ${(A\cup B)}^c$.

Assume that A flips first, then B, then C, then A, and so on.

A system is composed of $5$ components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector $(x_1, x_2, x_3, x_4, x_5)$, where $x_i$ is equal to $1$ if component $i$ is working and is equal to $0$ if component $i$ is failed.

(a) How many outcomes are in the sample space of this experiment?

(b) Suppose that the system will work if components $1$ and $2$ are both working, or if components $3$ and $4$ are both working, or if components $1$, $3$, and $5$ are all working. Let W be the event that the system will work. Specify all the outcomes in W.

(c) Let $A$ be the event that components $4$ and $5$ are both failed. How many outcomes are contained in the event $A$?

(d) Write out all the outcomes in the event $AW$.

A hospital administrator codes incoming patients suffering gunshot wounds according to whether they have insurance (coding $1$ if they do and $0$ if they do not) and according to their condition, which is rated as good (g), fair (f), or serious (s). Consider an experiment that consists of

the coding of such a patient.

(a) Give the sample space of this experiment.

(b) Let $A$ be the event that the patient is in serious condition. Specify the outcomes in $A$.

(c) Let $B$ be the event that the patient is uninsured. Specify the outcomes in $B$.

(d) Give all the outcomes in the event $B^c \cup A$.

Consider an experiment that consists of determining the type of job—either blue collar or white collar— and the political affiliation—Republican, Democratic, or Independent—of the 15 members of an adult soccer team.

How many outcomes are

(a) in the sample space?

(b) in the event that at least one of the team members is a blue-collar worker?

(c) in the event that none of the team members considers himself or herself an Independent?

Suppose that A and B are mutually exclusive events for which $P\left(A\right) = .3 \text{and} P\left(B\right) = .5$. What is the probability that

(a) either A or B occurs?

(b) A occurs but B does not?

(c) both A and B occur?

A retail establishment accepts either the American Express or the VISA credit card. A total of $24$ percent of its customers carry an American Express card, $61$ percent carry a VISA card, and $11$ percent carry both cards. What percentage of its customers carry a credit card that

the establishment will accept?

Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty percent wear a ring and $30$ percent wear a necklace. If one of the students is chosen randomly, what is the probability that this student is wearing

(a) a ring or a necklace?

(b) a ring and a necklace?

A total of $28$ percent of American males smoke cigarettes, $7$ percent smoke cigars, and $5$ percent smoke both cigars and cigarettes.

(a)What percentage of males smokes neither cigars nor cigarettes?

(b)What percentage smokes cigars but not cigarettes?

If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.  

Two cards are randomly selected from an ordinary playing deck. What is the probability that they form a blackjack? That is, 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?

Two symmetric dice have had two of their sides painted red, two painted black, one painted yellow, and the other

painted white. When this pair of dice are rolled, what is the probability that both dice land with the same color face up?

Suppose that you are playing blackjack against a dealer. In a freshly shuffled deck, what is the probability that neither you nor the dealer is dealt a blackjack?

Poker dice is played by simultaneously rolling $5$ dice. Show that

(a) P{no two alike}

(b) P{one pair}

(c) P{two pair}

(d) P{three alike}

(e) P{full house}

(f) P{four alike}

(g) P{five alike}

If a rook (castles) are randomly placed on chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

Two cards are randomly selected from an ordinary playing deck. What is the probability that it is a blackjack$?$ That is, what is the probability that one of the card is an ace and the other one is either a a ten, a jack, a queen or a king$?$

Two symmetric dice have had two of their sides painted red, two sides painted black, one painted yellow and the other painted white. When this pair of dice rolled, what is the probability that both dice land with same color face up$?$

Suppose that you are playing blackjack against a dealer. In a freshly shuffled deck, what is the probability that neither you nor the dealer is dealt a blackjack$?$

A certain town with a population $100,000$has $3$newspapers:$I, II,$ and $III.$ The proportions of townspeople who read these papers are as follows:

$I: 10$ percent $I$ and $II: 8$ percent$I$,$II$ and $III: 1$ percent

$II: 30$ percent$I$ and $III: 2$percent

$III: 5$percent$II$ and$III: 4$ percent

(The list tells us, for instance, that$8000$ people read newspapers $I$and$II$)

$\left(a\right)$ Find the number of people who read only one newspaper.

$\left(b\right)$ How many people read at least two newspapers?

$\left(c\right)$ If$I$ and $III$are morning papers and $II$is an evening paper, how many people read at least one-morning paper plus an evening paper?

$\left(d\right)$How many people do not read any newspapers?

$\left(e\right)$How many people read the only one-morning paper and one evening paper?

The following data were given in a study of a group of$1000$ subscribers to a certain magazine: In reference to the job, marital status, and education, there were $312$professionals, $470$married persons, $525$ college graduates, $42$professional college graduates, $147$ married college graduates, $86$ married professionals, and $25$married professional college graduates. Show that the numbers reported in the

the study must be incorrect.

Hint: Let $M, W,$and$G$ denote, respectively, the set of professionals, married persons, and college graduates. Assume that one of the $1000$persons is chosen at random, and use Proposition $4.4$to show that if the given numbers are correct, then$P(M \cup W \cup G) > 1$.

If it is assumed that all$\left(\begin{array}{c}52\\ 5\end{array}\right)$ poker hands are equally likely, what is the probability of being dealt

$\left(a\right)$ a flush? (A hand is said to be a flush if all $5$ cards are of the same suit.)

$\left(b\right)$ one pair? (This occurs when the cards have denominations $a, a, b, c, d,$ where $a, b, c,$ and$d$ are all distinct.)

$\left(c\right)$ two pairs? (This occurs when the cards have denominations $a, a, b, b, c,$ where $a, b,$ and $c$ are all distinct.)

$\left(d\right)$ three of a kind? (This occurs when the cards have denominations $a, a, a, b, c,$ where $a, b,$ and $c$ are all distinct.)

$\left(e\right)$ four of a kind? (This occurs when the cards have denominations$a, a, a, a, b$)

Poker dice are played by simultaneously rolling $5$dice. Show that

If $8$rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

A small community organization consists of $20$families, which $4$have one child, $8$ have two children, $5$have three children, $2$have four children, and$1$ have five children.

$\left(a\right)$ If one of these families is chosen at random, what is the probability it has$i$ children, $i = 1, 2, 3, 4, 5?$

$\left(b\right)$ If one of the children is randomly chosen, what is the probability that the child comes from a family having$i$ children,$i = 1, 2, 3, 4, 5?$

An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The

classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students who are in both Spanish and French, 4 who are in both Spanish and German, and 6 who are in both French and German. In addition, there are 2 students taking all 3 classes.

(a) If a student is chosen randomly, what is the probability that he or she is not in any of the language classes?

(b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class?

(c) If 2 students are chosen randomly, what is the probability that at least 1 is taking a language class?

A pair of dice is rolled until a sum of either $5$or $7$ appears. Find the probability that a $5$ occurs first.

 Hint: Let $E_n$ denote the event that a $5$ occurs on the nth roll and no $5 or 7$ occurs on the first $n-1$ rolls. Compute $P\left(E_n\right)$ and argue that $\sum _{n=1}^{\infty }P\left(E_n\right)$is the desired probability 

The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a$2,3,or 12$, the player loses; if the sum is either a $7$ or an $11$, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a $7$. If the $7$ comes first, the player loses, whereas if the initial outcome reoccurs before the $7$ appears, the player wins. Compute the probability of a player winning at craps.

 Hint: Let $E_i$denote the event that the initial outcome is $i$and the player wins. The desired probability is $\sum _{i=12}^{12}P\left(E_i\right)$. To compute $P\left(E_i\right)$, define the events $E_{i,n}$ to be the event that the initial sum is i and the player wins on the nth roll. Argue that  

$P\left(E_i\right)=\sum _{n=1}^{\infty }P\left(E_{i,n}\right)$

An urn contains $5$ red, $6$ blue, and $8$ green balls. If a set of $3$ balls is randomly selected, what is the probability that each of the balls will be

(a) of the same color? 

(b) of different colors? Repeat under the assumption that whenever a ball is selected, its color is noted and it is then replaced in the urn before the next selection. This is known as sampling with replacement .

The chess clubs of two schools consist of, respectively, $8 and 9$  players. Four members from each club are randomly chosen to participate in a contest between the two schools. The chosen players from one team are then randomly paired with those from the other team, and each pairing plays a game of chess. Suppose that Rebecca and her sister Elise are on the chess clubs at different schools. What is the probability that

(a) Rebecca and Elise will be paired? 

(b) Rebecca and Elise will be chosen to represent their schools but will not play each other?

 (c) either Rebecca or Elise will be chosen to represent her school? 

Consider the following technique for shuffling a deck of n cards: For any initial ordering of the cards, go through the deck one card at a time, and at each card, flip a fair coin. If the coin comes up heads, then leave the card where it is; if the coin comes up tails, then move that card to the end of the deck. After the coin has been flipped n times, say that one round has been completed. For instance, if $n = 4$the initial ordering is$1, 2, 3, 4,$ then if the successive flips result in the outcome $h, t, t, h,$then the ordering at the end of the round is $1, 4, 2, 3.$ Assuming that all possible outcomes of the sequence of $n$coin flips are equally likely, what is the probability that the ordering after one round is the same as the initial ordering?

A pair of fair dice is rolled. What is the probability that the second die lands on a higher value than does the first?

If two dice are rolled, what is the probability that the sum of the upturned faces equals$i?$ Find it for$i =2, 3, ... , 11, 12.$

An urn contains $3$red and $7$black balls. Players$A$$\&$ $B$withdraw balls from the urn consecutively until a red ball is selected. Find the probability that $A$selects the red

ball. ($A$draws the first ball, then$B$, and so on. There is no replacement of the balls drawn.)

An urn contains $n$white and $m$black balls, where$n$and$m$are positive numbers.

$\left(a\right)$ If two balls are randomly withdrawn, what is the probability that they are the same color?

$\left(b\right)$ If a ball is randomly withdrawn and then replaced before the second one is drawn, what is the probability that the withdrawn balls are the same color?

$\left(c\right)$ Show that the probability in part $\left(b\right)$ is always larger than the one in part $\left(a\right)$.

A $3$-personal basketball team consists of a guard, a forward, and a center.

$\left(a\right)$If a person is chosen at random from each of three different such teams, what is the probability of selecting a complete team?

$\left(b\right)$What is the probability that all $3$players selected play the same position?

A group of individuals containing $b$boys and $g$girls is lined up in random order; that is, each of the$(b + g)!$ permutations is assumed to be equally likely. What is the probability that the person in the ith position, $1 \le i \le b + g,$ is a girl?

A forest contains$20$ elk, which $5$are captured,  tagged, and then released. A certain time later,$4$ the $20$elk are captured. What is the probability that $2$these $4$have been tagged? What assumptions are you making?

The second Earl of Yarborough is reported to have bet at odds $1000$-$1$that a bridge hand of$13$ cards would contain at least one card that is ten or higher. (By ten or higher we mean that a card is either a ten, a jack, a queen, a king, or an ace.) Nowadays, we call a hand that has no cards higher than$9$ a Yarborough. What is the probability that a randomly selected bridge hand is a Yarborough?

If a die is rolled 4 times, what is the probability that 6 comes up at least once? 

If there are $12$strangers in a room, what is the probability that no two of them celebrate their birthday in the same month?

Two cards are chosen at random from a deck of $52$playing cards. What is the probability that they

(a) are both aces?

(b) have the same value?

An instructor gives her class a set of $10$problems with the information that the final exam will consist of a random selection of $5$them. If a student has figured out how to do $7$the problems, what is the probability that he or she will answer correctly

$\left(a\right)$all $5$problems?

$\left(b\right)$ at least$4$ of the problems?

There are $5$hotels in a certain town. If $3$people check

into hotels in a day, what is the probability that they each check into a different hotel? What assumptions are you making?

If a die is rolled $4$times, what is the probability that $6$

comes up at least once?

There are$n$ socks, $3$which are red, in the drawer. What is the value of n if, when $2$the socks are chosen randomly, the probability that they are both red is$\frac{1}{2}$?

Two dice are thrown $n$times in succession. Compute

the probability that a double $6$appears at least once. How large need $n$be to make this probability at least$\frac{1}{2}$?

$\left(a\right)$ If $N$people, including $A$and$B$, are randomly arranged in a line, what is the probability that $A$and $B$are next to each other?

$\left(b\right)$ What would the probability be if the people were randomly arranged in a circle?