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A First Course in Probability

Sheldon Ross

Chapter 4

Random Variables - all with Video Answers

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Chapter Questions

04:43

Problem 1

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $\$ 2$ for each black ball selected and we lose $\$ 1$ 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?

JS
Johan Sannemo
Numerade Educator
01:44

Problem 2

Two fair dice are rolled. Let $X$ equal the product of the 2 dice. Compute $P\{X=i\}$ for $i=1, \ldots, 36$

Nafis Fuad
Nafis Fuad
Numerade Educator
02:54

Problem 3

Three dice are rolled. By assuming that each of the $6^{3}=216$ possible outcomes is equally likely, find the probabilities attached to the possible values that $X$ can take on, where $X$ is the sum of the 3 dice.

Sanchit Jain
Sanchit Jain
Numerade Educator
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Problem 4

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all $10 !$ possible rankings are equally likely. Let $X$ denote the highest ranking achieved by a woman. (For instance, $X=1$ if the top-ranked person is female.) Find $\mathrm{P}\{\mathrm{X}=\mathrm{i}\}, i=1,2,3, \ldots, 8,9,10$.

Emily Himsel
Emily Himsel
Numerade Educator
01:42

Problem 5

Let $X$ represent the difference between the number of heads and the number of tails obtained when a coin is tossed $n$ times. What are the possible values of $X ?$

Manisha Sarker
Manisha Sarker
Numerade Educator
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Problem 6

In Problem $4.5,$ for $n=3,$ if the coin is assumed fair, what are the probabilities associated with the values that $X$ can take on?

Jason Gerber
Jason Gerber
Numerade Educator
01:23

Problem 7

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on:
(a) the maximum value to appear in the two rolls;
(b) the minimum value to appear in the two rolls;
(c) the sum of the two rolls;
(d) the value of the first roll minus the value of the second roll?

Raj Bala
Raj Bala
Numerade Educator
08:26

Problem 8

If the die in Problem 4.7 is assumed fair, calculate the probabilities associated with the random variables in parts
(a) through (d).

Ahmad Reda
Ahmad Reda
Numerade Educator
03:25

Problem 9

Repeat Example $1 \mathrm{c}$ when the balls are selected with replacement.

Amany Waheeb
Amany Waheeb
Numerade Educator
05:44

Problem 10

Let $X$ be the winnings of a gambler. Let $p(i)=$ $P(X=i)$ and suppose that
$$\begin{array}{l}
p(0)=1 / 3 ; p(1)=p(-1)=13 / 55 \\
p(2)=p(-2)=1 / 11 ; p(3)=p(-3)=1 / 165
\end{array}$$
Compute the conditional probability that the gambler wins $i, i=1,2,3,$ given that he wins a positive amount.

Ramon Kryzhan
Ramon Kryzhan
Numerade Educator
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Problem 11

(a) An integer $N$ is to be selected at random from $\left\{1,2, \ldots,(10)^{3}\right\}$ in the sense that each integer has the same probability of being selected. What is the probability that $N$ will be divisible by $3 ?$ by $5 ?$ by $7 ?$ by $15 ?$ by $105 ?$ How would your answer change if $(10)^{3}$ is replaced by $(10)^{k}$ as $k$ became larger and larger?
(b) An important function in number theory-one whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis - is the Möbius function $\mu(n)$ defined for all positive integral values $n$ as follows: Factor $n$ into its prime factors. If there is a repeated prime factor, as in $12=2 \cdot 2 \cdot 3$ or $49=7 \cdot 7,$ then $\mu(n)$ is defined to equal 0. Now let $N$ be chosen at random from $\left\{1,2, \ldots(10)^{k}\right\}$ where $k$ is large. Determine $P\{\mu(N)=0\}$ as $k \rightarrow \infty$ Hint: To compute $P\{\mu(N) \neq 0\},$ use the identity
$$\prod_{i=1}^{\infty} \frac{P_{i}^{2}-1}{P_{i}^{2}}=\left(\frac{3}{4}\right)\left(\frac{8}{9}\right)\left(\frac{24}{25}\right)\left(\frac{48}{49}\right) \cdots=\frac{6}{\pi^{2}}$$
where $P_{i}$ is the $i$ th-smallest prime. (The number 1 is not a prime.)

Victor Salazar
Victor Salazar
Numerade Educator
00:41

Problem 12

In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by $X$ the amount of money he wins in a single game of Two-Finger Morra.
(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of $X$ and what are their associated probabilities?
(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1 or 2 fingers, what are the possible values of $X$ and their associated probabilities?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:14

Problem 13

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability $.3,$ and his second will lead independently to a sale with probability.6. Any sale made is equally likely to be either for the deluxe model, which costs $\$ 1000,$ or the standard model, which costs $\$ 500 .$ Determine the probability mass function of $X$, the total dollar value of all sales.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
04:21

Problem 14

Five distinct numbers are randomly distributed to players numbered 1 through $5 .$ Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares her number with that of player $3,$ and so on. Let $X$ denote the number of times player 1 is a winner. Find $P\{X=i\}, i=0,1,2,3,4$.

George Prestidge
George Prestidge
Numerade Educator
03:04

Problem 15

The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst won-lost records during the year. A total of 66 balls are placed in an urn. Each of these balls is inscribed with the name of a team: Eleven have the name of the team with the worst record, 10 have the name of the team with the secondworst record, 9 have the name of the team with the thirdworst record, and so on (with 1 ball having the name of the team with the 11 th-worst record). A ball is then chosen at random, and the team whose name is on the ball is given the first pick in the draft of players about to enter the league. Another ball is then chosen, and if it "belongs" to a team different from the one that received the first draft pick, then the team to which it belongs receives the second draft pick. (If the ball belongs to the team receiving the first pick, then it is discarded and another one is chosen; this continues until the ball of another team is chosen.) Finally, another ball is chosen, and the team named on the ball (provided that it is different from the previous two teams) receives the third draft pick. The remaining draft picks 4 through 11 are then awarded to the 8 teams that did not "win the lottery," in inverse order of their won-lost records. For instance, if the team with the worst record did not receive any of the 3 lottery picks, then that team would receive the fourth draft pick. Let $X$ denote the draft pick of the team with the worst record. Find the probability mass function of $X$.

Jeff Vermeire
Jeff Vermeire
Numerade Educator
08:42

Problem 16

In Problem $4.15,$ let team number 1 be the team with the worst record, let team number 2 be the team with the second-worst record, and so on. Let $Y_{i}$ denote the team that gets draft pick number $i$. (Thus, $Y_{1}=3$ if the first ball chosen belongs to team number $3 .$ Find the probability mass function of (a) $Y_{1},$ (b) $Y_{2},$ and (c) $Y_{3}$.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:46

Problem 17

Suppose that the distribution function of $X$ is given by
$$F(b)=\left\{\begin{array}{ll}
0 & b < 0 \\
\frac{b}{4} & 0 \leq b < 1 \\
\frac{1}{2}+\frac{b-1}{4} & 1 \leq b < 2 \\
\frac{11}{12} & 2 \leq b < 3 \\
1 & 3 \leq b
\end{array}\right.$$
(a) Find $P\{X=i\}, i=1,2,3$
(b) Find $P\left\{\frac{1}{2} < X < \frac{3}{2}\right\}$

Linh Vu
Linh Vu
Numerade Educator
04:31

Problem 18

Four independent flips of a fair coin are made. Let $X$ denote the number of heads obtained. Plot the probability mass function of the random variable $X-2$.

Willis James
Willis James
Numerade Educator
02:24

Problem 19

If the distribution function of $X$ is given by
$$F(b)=\left\{\begin{array}{ll}
0 & b<0 \\
\frac{1}{2} & 0 \leq b < 1 \\
\frac{3}{5} & 1 \leq b < 2 \\
\frac{4}{5} & 2 \leq b < 3 \\
\frac{9}{10} & 3 \leq b < 3.5 \\
1 & b \geq 3.5
\end{array}\right.$$
calculate the probability mass function of $X$

Amany Waheeb
Amany Waheeb
Numerade Educator
02:33

Problem 20

A gambling book recommends the following "winning strategy" for the game of roulette: Bet $\$ 1$ on red. If red appears (which has probability $\frac{18}{38}$ ), then take the $\$ 1$ profit and quit. If red does not appear and you lose this bet (which has probability $\frac{20}{38}$ of occurring), make additional $\$ 1$ bets on red on each of the next two spins of the roulette wheel and then quit. Let $X$ denote your winnings when you quit.
(a) Find $P\{X > 0\}$
(b) Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer!
(c) Find $E[X]$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:34

Problem 21

Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, $40,33,25,$ and 50 students. One of the students is randomly selected. Let $X$ denote the number of students who were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let $Y$ denote the number of students on her bus.
(a) Which of $E[X]$ or $E[Y]$ do you think is larger? Why?
(b) Compute $E[X]$ and $E[Y]$

Prashant Bana
Prashant Bana
Numerade Educator
08:42

Problem 22

Suppose that two teams play a series of games that ends when one of them has won $i$ games. Suppose that each game played is, independently, won by team $A$ with probability $p .$ Find the expected number of games that are played when (a) $i=2$ and (b) $i=3 .$ Also, show in both cases that this number is maximized when $p=\frac{1}{2}$.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:15

Problem 23

You have $\$ 1000,$ and a certain commodity presently sells for $\$ 2$ per ounce. Suppose that after one week the commodity will sell for either $\$ 1$ or $\$ 4$ an ounce, with these two possibilities being equally likely.
(a) If your objective is to maximize the expected amount of money that you possess at the end of the week, what strategy should you employ?
(b) If your objective is to maximize the expected amount of the commodity that you possess at the end of the week, what strategy should you employ?

Tony Ni
Tony Ni
Numerade Educator
01:29

Problem 24

$A$ and $B$ play the following game: $A$ writes down either number 1 or number $2,$ and $B$ must guess which one. If the number that $A$ has written down is $i$ and $B$ has guessed correctly, $B$ receives $i$ units from $A$. If $B$ makes a wrong guess, $B$ pays $\frac{3}{4}$ unit to $A .$ If $B$ randomizes his decision by guessing 1 with probability $p$ and 2 with probability $1-p,$ determine his expected gain if (a) $A$ has written down number 1 and (b) $A$ has written down number 2 What value of $p$ maximizes the minimum possible value of $B$ 's expected gain, and what is this maximin value? (Note that $B$ 's expected gain depends not only on $p,$ but also on what $A$ does.) Consider now player $A$. Suppose that she also randomizes her decision, writing down number 1 with probability
q. What is $A$ 's expected loss if (c) $B$ chooses number 1 and (d) $B$ chooses number $2 ?$ What value of $q$ minimizes $A$ 's maximum expected loss? Show that the minimum of $A$ 's maximum expected loss is equal to the maximum of $B$ 's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player $B$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:37

Problem 25

Two coins are to be flipped. The first coin will land on heads with probability $.6,$ the second with probability $.7 .$ Assume that the results of the flips are independent, and let $X$ equal the total number of heads that result.
(a) Find $P\{X=1\}$
(b) Determine $E[X]$

Amany Waheeb
Amany Waheeb
Numerade Educator
01:04

Problem 26

One of the numbers 1 through 10 is randomly chosen. You are to try to guess the number chosen by asking questions with "yes-no" answers. Compute the expected number of questions you will need to ask in each of the following two cases:
(a) Your $i$ th question is to be "Is it i?" $i=$ 1,2,3,4,5,6,7,8,9,10
(b) With each question you try to eliminate one-half of the remaining numbers, as nearly as possible.

Jeff Vermeire
Jeff Vermeire
Numerade Educator
06:35

Problem 27

An insurance company writes a policy to the effect that an amount of money $A$ must be paid if some event $E$ occurs within a year. If the company estimates that $E$ will occur within a year with probability $p,$ what should it charge the customer in order that its expected profit will be 10 percent of $A ?$

Robin Corrigan
Robin Corrigan
Numerade Educator
01:42

Problem 28

A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.

Goutam Chand
Goutam Chand
Numerade Educator
01:11

Problem 29

There are two possible causes for a breakdown of a machine. To check the first possibility would cost $C_{1}$ dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of $R_{1}$ dollars. Similarly, there are costs $C_{2}$ and $R_{2}$ associated with the second possibility. Let $p$ and $1-p$ denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on $p, C_{i}, R_{i}, i=1,2$ should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order? Note: If the first check is negative, we must still check the other possibility.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
00:38

Problem 30

A person tosses a fair coin until a tail appears for the first time. If the tail appears on the $n$ th flip, the person wins $2^{n}$ dollars. Let $X$ denote the player's winnings. Show that $E[X]=+\infty .$ This problem is known as the St. Petersburg paradox.
(a) Would you be willing to pay $\$ 1$ million to play this game once?
(b) Would you be willing to pay $\$ 1$ million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
03:27

Problem 31

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p,$ then he or she will receive a score of $1-(1-p)^{2} \quad$ if it does rain $1-p^{2}$ if it does not rain We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability $p^{*},$ what value of $p$ should he or she assert so as to maximize the expected score?

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
00:46

Problem 32

To determine whether they have a certain disease, 100 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to place the people into groups of $10 .$ The blood samples of the 10 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 10 people, whereas if the test is positive, each of the 10 people will also be individually tested and, in all, 11 tests will be made on this group. Assume that the probability that a person has the disease is .1 for all people, independently of one another, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)

Hunza Gilgit
Hunza Gilgit
Numerade Educator
07:21

Problem 33

A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with $n=10, p=\frac{1}{3},$ approximately how many papers should he purchase so as to maximize his expected profit?

Ahmad Reda
Ahmad Reda
Numerade Educator
01:05

Problem 34

In Example $4 \mathrm{b},$ suppose that the department store incurs an additional cost of $c$ for each unit of unmet demand. (This type of cost is often referred to as a goodwill cost because the store loses the goodwill of those customers whose demands it cannot meet.) Compute the expected profit when the store stocks $s$ units, and determine the value of $s$ that maximizes the expected profit.

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 35

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $\$ 1.10 ;$ if they are different colors, then you win $-\$ 1.00 .$ (That is, you lose $\$ 1.00 .$ ) Calculate
(a) the expected value of the amount you win;
(b) the variance of the amount you win.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:20

Problem 36

Consider Problem 4.22 with $i=2 .$ Find the variance of the number of games played, and show that this number is maximized when $p=\frac{1}{2}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
01:22

Problem 37

Find $\operatorname{Var}(X)$ and $\operatorname{Var}(Y)$ for $X$ and $Y$ as given in Problem 4.21.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
04:05

Problem 38

If $E[X]=1$ and $\operatorname{Var}(X)=5,$ find
(a) $E\left[(2+X)^{2}\right]$
(b) $\operatorname{Var}(4+3 X)$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:12

Problem 39

A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is replaced and another ball is drawn. This process goes on indefinitely. What is the probability that of the first 4 balls drawn, exactly 2 are white?

Wendi Zhao
Wendi Zhao
Numerade Educator
01:56

Problem 40

On a multiple-choice exam with 3 possible answers for each of the 5 questions, what is the probability that a student will get 4 or more correct answers just by guessing?

Steven Clarke
Steven Clarke
Numerade Educator
00:53

Problem 41

A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct. What is the probability that he would have done at least this well if he did not have ESP?

Nick Johnson
Nick Johnson
Numerade Educator
08:57

Problem 42

$A$ and $B$ will take the same 10 -question examination. Each question will be answered correctly by $A$ with probability.7, independently of her results on other questions. Each question will be answered correctly by $B$ with probability $.4,$ independently both of her results on the other questions and on the performance of $A$
(a) Find the expected number of questions that are answered correctly by both $A$ and $B$.
(b) Find the variance of the number of questions that are answered correctly by either $A$ or $B$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
02:22

Problem 43

A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0 and 11111 instead of $1 .$ If the receiver of the message uses "majority" decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?

Maxime Rossetti
Maxime Rossetti
Numerade Educator
08:29

Problem 44

A satellite system consists of $n$ components and functions on any given day if at least $k$ of the $n$ components function on that day. On a rainy day, each of the components independently functions with probability $p_{1},$ whereas on a dry day, each independently functions with probability $p_{2} .$ If the probability of rain tomorrow is $\alpha,$ what is the probability that the satellite system will function?

Robin Corrigan
Robin Corrigan
Numerade Educator
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Problem 45

A student is getting ready to take an important oral examination and is concerned about the possibility of having an "on" day or an "off" day. He figures that if he has an on day, then each of his examiners will pass him, independently of one another, with probability $.8,$ whereas if he has an off day, this probability will be reduced to .4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?

Donna Densmore
Donna Densmore
Numerade Educator
11:28

Problem 46

Suppose that it takes at least 9 votes from a 12 member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is .2 whereas the probability that the juror votes an innocent person guilty is .1. If each juror acts independently and if 65 percent of the defendants are guilty, find the probability that the jury renders a correct decision. What percentage of defendants is convicted?

Ahmad Reda
Ahmad Reda
Numerade Educator
13:23

Problem 47

In some military courts, 9 judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability. $7,$ whereas when the defendant is, in fact, innocent, this probability drops to .3.
(a) What is the probability that a guilty defendant is declared guilty when there are (i) $9,$ (ii) $8,$ and (iii) 7 judges?
(b) Repeat part (a) for an innocent defendant.
(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is 60 percent certain that the client is guilty?

Robin Corrigan
Robin Corrigan
Numerade Educator
08:15

Problem 48

It is known that diskettes produced by a certain company will be defective with probability.01, independently of one another. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of the 10 diskettes in the package will be defective. The guarantee is that the customer can return the entire package of diskettes if he or she finds more than 1 defective diskette in it. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them?

Tatiana Graham
Tatiana Graham
Numerade Educator
04:30

Problem 49

When coin 1 is flipped, it lands on heads with probability $.4 ;$ when coin 2 is flipped, it lands on heads with probability $.7 .$ One of these coins is randomly chosen and flipped 10 times.
(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
(b) Given that the first of these 10 flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?

Philomena Marfo
Philomena Marfo
Numerade Educator
01:53

Problem 50

Suppose that a biased coin that lands on heads with probability $p$ is flipped 10 times. Given that a total of 6 heads results, find the conditional probability that the first 3 outcomes are
(a) $h, t, t$ (meaning that the first flip results in heads, the second in tails, and the third in tails);
(b) $t, h, t$

Amany Waheeb
Amany Waheeb
Numerade Educator
03:00

Problem 51

The expected number of typographical errors on a page of a certain magazine is .2. What is the probability that the next page you read contains (a) 0 and (b) 2 or more typographical errors? Explain your reasoning!

Amany Waheeb
Amany Waheeb
Numerade Educator
06:35

Problem 52

The monthly worldwide average number of airplane crashes of commercial airlines is $3.5 .$ What is the probability that there will be
(a) at least 2 such accidents in the next month;
(b) at most 1 accident in the next month? Explain vour reasoning!

Amany Waheeb
Amany Waheeb
Numerade Educator
05:49

Problem 53

Approximately 80,000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couples,
(a) both partners were born on April 30 ;
(b) both partners celebrated their birthday on the same day of the year. State your assumptions.

Ahmad Reda
Ahmad Reda
Numerade Educator
00:46

Problem 54

Suppose that the average number of cars abandoned weekly on a certain highway is $2.2 .$ Approximate the probability that there will be
(a) no abandoned cars in the next week;
(b) at least 2 abandoned cars in the next week.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:16

Problem 55

A certain typing agency employs 2 typists. The average number of errors per article is 3 when typed by the first typist and 4.2 when typed by the second. If your article is equally likely to be typed by either typist, approximate the probability that it will have no errors.

Prashant Bana
Prashant Bana
Numerade Educator
00:52

Problem 56

How many people are needed so that the probability that at least one of them has the same birthday as you is greater than $\frac{1}{2} ?$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:02

Problem 57

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter $\lambda=3$
(a) Find the probability that 3 or more accidents occur today.
(b) Repeat part (a) under the assumption that at least 1 accident occurs today.

Christopher Stanley
Christopher Stanley
Numerade Educator
04:55

Problem 58

Compare the Poisson approximation with the correct binomial probability for the following cases:
(a) $P\{X=2\}$ when $n=8, p=.1$
(b) $P\{X=9\}$ when $n=10, p=.95$
(c) $P\{X=0\}$ when $n=10, p=.1$
(d) $P\{X=4\}$ when $n=9, p=.2$

Neel Faucher
Neel Faucher
Numerade Educator
02:13

Problem 59

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is $\frac{1}{100},$ what is the (approximate) probability that you will win a prize
(a) at least once?
(b) exactly once?
(c) at least twice?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:15

Problem 60

The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter $\lambda=5 .$ Suppose that a new wonder drug (based on large quantities of vitamin $\mathrm{C}$ ) has just been marketed that reduces the Poisson parameter to $\lambda=3$ for 75 percent of the population. For the other 25 percent of the population, the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her?

Barsha Rana
Barsha Rana
Numerade Educator
05:44

Problem 61

The probability of being dealt a full house in a hand of poker is approximately.0014. Find an approximation for the probability that in 1000 hands of poker, you will be dealt at least 2 full houses.

Chris Trentman
Chris Trentman
Numerade Educator
03:15

Problem 62

Consider $n$ independent trials, each of which results in one of the outcomes $1, \ldots, k$ with respective probabilities $p_{1}, \ldots, p_{k}, \quad \sum_{i=1}^{k} p_{i}=1 .$ Show that if all the $p_{i}$ are small, then the probability that no trial outcome occurs more than once is approximately equal to $\exp (-n(n-1)$ $\left.\sum_{i} p_{i}^{2} / 2\right)$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:52

Problem 63

People enter a gambling casino at a rate of 1 every 2 minutes.
(a) What is the probability that no one enters between 12: 00 and $12: 05 ?$
(b) What is the probability that at least 4 people enter the casino during that time?

Dominador Tan
Dominador Tan
Numerade Educator
02:28

Problem 64

The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month.
(a) Find the probability that in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month.
(b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides?
(c) Counting the present month as month number $1,$ what is the probability that the first month to have 8 or more suicides will be month number $i, i \geq 1 ?$ What assumptions are you making?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:11

Problem 65

Each of 500 soldiers in an army company independently has a certain disease with probability $1 / 10^{3} .$ This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested.
(a) What is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)? Suppose now that the blood test yields a positive result.
(b) What is the probability, under this circumstance, that more than one person has the disease? Now, suppose one of the 500 people is Jones, who knows that he has the disease.
(c) What does Jones think is the probability that more than one person has the disease? Because the pooled test was positive, the authorities have decided to test each individual separately. The first $i-1$ of these tests were negative, and the $i$ th one-which was on Jones-was positive.
(d) Given the preceding scenario, what is the probability, as a function of $i,$ that any of the remaining people have the disease?

Foster Wisusik
Foster Wisusik
Numerade Educator
02:13

Problem 66

A total of $2 n$ people, consisting of $n$ married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let $C_{i}$ denote the event that the members of couple $i$ are seated next to each other, $i=1, \ldots, n$
(a) Find $P\left(C_{i}\right)$
(b) For $j \neq i,$ find $P\left(C_{j} | C_{i}\right)$
(c) Approximate the probability, for $n$ large, that there are no married couples who are seated next to each other.

Aman Gupta
Aman Gupta
Numerade Educator
03:25

Problem 67

Repeat the preceding problem when the seating is random but subject to the constraint that the men and women alternate.

Shoukat Ali
Shoukat Ali
Other Schools
01:57

Problem 68

In response to an attack of 10 missiles, 500 antiballistic missiles are launched. The missile targets of the antiballistic missiles are independent, and each antiballstic missile is equally likely to go towards any of the target missiles. If each antiballistic missile independently hits its target with probability $.1,$ use the Poisson paradigm to approximate the probability that all missiles are hit.

Christopher Stanley
Christopher Stanley
Numerade Educator
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Problem 69

A fair coin is flipped 10 times. Find the probability that there is a string of 4 consecutive heads by
(a) using the formula derived in the text;
(b) using the recursive equations derived in the text.
(c) Compare your answer with that given by the Poisson approximation.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:53

Problem 70

At time $0,$ a coin that comes up heads with probability $p$ is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate $\lambda,$ the coin is picked up and flipped. (Between these times, the coin remains on the ground.) What is the probability that the coin is on its head side at time $t ?$ Hint: What would be the conditional probability if there were no additional flips by time $t,$ and what would it be if there were additional flips by time $t ?$

Amany Waheeb
Amany Waheeb
Numerade Educator
00:54

Problem 71

Consider a roulette wheel consisting of 38 numbers 1 through $36,0,$ and double $0 .$ If Smith always bets that the outcome will be one of the numbers 1 through $12,$ what is the probability that
(a) Smith will lose his first 5 bets;
(b) his first win will occur on his fourth bet?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
08:42

Problem 72

Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability $.6,$ independently of the outcomes of the other games. Find the probability, for $i=4,5,6,7,$ that the stronger team wins the series in exactly $i$ games. Compare the probability that the stronger team wins with the probability that it would win a 2-outof-3 series.

Ahmad Reda
Ahmad Reda
Numerade Educator
08:42

Problem 73

Suppose in Problem 4.72 that the two teams are evenly matched and each has probability $\frac{1}{2}$ of winning each game. Find the expected number of games played.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:04

Problem 74

An interviewer is given a list of people she can interview. If the interviewer needs to interview 5 people, and if each person (independently) agrees to be interviewed with probability $\frac{2}{3},$ what is the probability that her list of people will enable her to obtain her necessary number of interviews if the list consists of (a) 5 people and (b) 8 people? For part (b), what is the probability that the interviewer will speak to exactly (c) 6 people and (d) 7 people on the list?

Robin Corrigan
Robin Corrigan
Numerade Educator
04:31

Problem 75

A fair coin is continually flipped until heads appears for the 10 th time. Let $X$ denote the number of tails that occur. Compute the probability mass function of $X .$

Willis James
Willis James
Numerade Educator
04:08

Problem 76

Solve the Banach match problem (Example 8e) when the left-hand matchbox originally contained $N_{1}$ matches and the right-hand box contained $N_{2}$ matches.

Joy Carpio
Joy Carpio
Numerade Educator
02:11

Problem 77

In the Banach matchbox problem, find the probability that at the moment when the first box is emptied (as opposed to being found empty), the other box contains exactly $k$ matches.

Bryan Lynn
Bryan Lynn
Numerade Educator
02:12

Problem 78

An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly $n$ selections?

Wendi Zhao
Wendi Zhao
Numerade Educator
04:59

Problem 79

Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If $X$ is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) $P\{X=0\}$ and (b) $P\{X > 2\}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
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Problem 80

A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 through 80. A player can select from 1 to 15 numbers; a win occurs if some fraction of the player's chosen subset matches any of the 20 numbers drawn by the house. The payoff is a function of the number of elements in the player's selection and the number of matches. For instance, if the player selects only 1 number, then he or she wins if this number is among the set of $20,$ and the payoff is $\$ 2.20$ won for every dollar bet. (As the player's probability of winning in this case is $\frac{1}{4},$ it is clear that the "fair" payoff should be $\$ 3$ won for every $\$ 1$ bet.) When the player selects 2 numbers, a payoff (of odds) of $\$ 12$ won for every $\$ 1$ bet is made when both numbers are among the $20 .$
(a) What would be the fair payoff in this case? Let $P_{n, k}$ denote the probability that exactly $k$ of the $n$ numbers chosen by the player are among the 20 selected by the house.
(b) Compute $P_{n, k}$
(c) The most typical wager at Keno consists of selecting 10 numbers. For such a bet, the casino pays off as shown in the following table. Compute the expected payoff:

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:16

Problem 81

In Example $8 \mathrm{i},$ what percentage of $i$ defective lots does the purchaser reject? Find it for $i=1,4 .$ Given that a lot is rejected, what is the conditional probability that it contained 4 defective components?

Manisha Sarker
Manisha Sarker
Numerade Educator
01:46

Problem 82

A purchaser of transistors buys them in lots of $20 .$ It is his policy to randomly inspect 4 components from a lot and to accept the lot only if all 4 are nondefective. If each component in a lot is, independently, defective with probability .1, what proportion of lots is rejected?

Hailey Tomashek
Hailey Tomashek
Numerade Educator
07:16

Problem 83

There are three highways in the county. The number of daily accidents that occur on these highways are Poisson random variables with respective parameters. $3, .5,$ and $.7 .$ Find the expected number of accidents that will happen on any of these highways today.

Danielle Ashley
Danielle Ashley
Numerade Educator
01:29

Problem 84

Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box $i$ with probability
$p_{i}, \sum_{i=1}^{5} p_{i}=1$
(a) Find the expected number of boxes that do not have any balls.
(b) Find the expected number of boxes that have exactly 1 ball.

Aman Gupta
Aman Gupta
Numerade Educator
01:24

Problem 85

There are $k$ types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type $i$ with probability $p_{i}, \quad \sum_{i=1}^{k} p_{i}=1$ If $n$ coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of $n$ coupons.)

Hoan Nguyen
Hoan Nguyen
Numerade Educator