Ch 4 Quizlet - Random Variables

4.62 If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1/100 , what is the (approximate) probability that you will win a prize (c) at least twice?

.09

4.74 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 (b) his first win will occur on his fourth bet?

.101

4.55 The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probabil- ity that there will be (b) at most 1 accident in the next month? Explain your reasoning!

.136

4.74 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;

.15

4.50 When coin 1 is flipped, it lands on heads with prob- ability .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?

.155

4.50 When coin 1 is flipped, it lands on heads with prob- ability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times. (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?

.197

4.62 If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1 100 , what is the (approximate) probability that you will win a prize (b) exactly once?

.30

4.62 If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1/100 , what is the (approximate) probability that you will win a prize (a) at least once?

.39

4.54 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 Explain your reasoning!

.82

4.55 The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probabil- ity that there will be (a) at least 2 such accidents in the next month;

.86

4.49 It is known that diskettes produced by a certain com- pany will be defective with probability .01, independently of one another. The company sells the diskettes in pack- ages 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 defec- tive diskette in it. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them?

P=.01278

PMF of X if X is discrete rv

Probability mass function A random variable whose set of possible values is either finite or countably infinite is called discrete. If X is a discrete random variable, then the function p(x) = P{X = x} is called the probability mass function of X.

Expected value and variance of a Poisson rv

The mean and variance of a Poisson random variable are both equal to its parameter λ.

4.69 A total of 2n people, consisting of n married cou- ples, are randomly seated (all possible orderings being equally likely) at a round table. Let Ci denote the event that the members of couple i are seated next to each other, i = 1, ... , n. (a) Find P(Ci). (b) For j is unequal to i, find P(Cj|Ci). (c) Approximate the probability, for n large, that there are no married couples who are seated next to each other.

a) 2/(2n-1) b) 2/(2n-2) c) e^-1

4.35 A box contains 5 red and 5 blue marbles. Two mar- bles 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.

a) E[Y]= -1/15 b) var(Y)= 49/45

4.56 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;

.45

4.82 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}.

a) P(X=0)= .522 b) P(X>=2) = .0126

4.86 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.

1.5

4.41 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?

11/243

4.42 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 cor- rect. What is the probability that he would have done at least this well if he did not have ESP?

176/1024

4.59 How many people are needed so that the probability that at least one of them has the same birthday as you is greater than 1/2 ?

253

4.85 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?

=.3439

4.64 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.

=.408

4.71 In response to an attack of 10 missiles, 500 antiballis- tic missiles are launched. The missile targets of the antibal- listic 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.

=.935

4.14 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.

P(X=0) = 1/2 P(X=1)= 1/6 P(X=2)= 1/12 P(X=3)= 1/20 P(X=4)= 1/5

standard deviation of a random variable X

the square root of the variance

the expected value of a sum of random variables is equal to...

the sum of their expected values.

4.39 If E[X] = 1 and Var(X) = 5, find (a) E[(2 + X)2]; (b) Var(4 + 3X).

E[(2+X)^2] = 14 var(4+3X)= 45

4.56 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, (b) both partners celebrated their birthday on the same day of the year.

about 1

4.54 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 (b) 2 or more typographical errors? Explain your reasoning!

.018

4.1 Two balls are chosen randomly from an urn contain- ing 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?

28/91, 16/91, 1/91, 32/91, 8/91, 6/91

4.5 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? 4.6 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?

4.5// Possible values of X are {n −2t ∣ t ∈ {0,1,...,n}} 4.6//If we toss the coin for n = 3 times, the possible values of Heads is k = 0, 1, 2, 3. If there was k Heads, the difference between the number of Heads and Tails is k−(3−k) = 2k−3. We plug in k= 0, 1, 2, 3 to get x, which are the values X can take on. Hence, X ∈ {−3,−1,1,3}. So, the required probabilities are P(X=-3) = P(no heads) = P(TTT) = (1/2)^3 P(X=-1) = P(1 Head)=P(HTT,THT,TTH)= 3⋅(1/2)^3 P(X=1) = P(2 heads)= P(HHT, THH, HTH) = 3⋅(1/2)^3 P(X=3) = P(no heads) = P(TTT) = (1/2)^3 Observe the connection between the required random variable and the number of Heads that have fallen

If X is a random variable, then the function F(x) is defined by ______ which is called the distribution function of X.

F(x) = P{X </= x} All probabilities concerning X can be stated in terms of F.

What is a random variable?

A real-valued function defined on the outcome of a probability experiment (is value depends on an experiment). Random variables are represented using letters. a variable whose value is unknown or a function that assigns values to each of an experiment's outcomes. A random variable can be either discrete (having specific values) or continuous (any value in a continuous range). For example, suppose X represents the number of 6's that can be obtained by rolling a fair die 3 times. The notation P(X=x) denotes the probability that the number of 6's that can be rolled obtained when a die is rolled 3 times (which we have defined as X) is x. Here, x can take on the values of 0, 1 , 2 and 3. --> P(X=0), P(X=1), P(X=2), P(X=3)

4.32 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 sam- ples of the 10 people in each group will be pooled and analyzed together. If the test is negative, one test will suf- fice 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 prob- ability 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.)

E[X] = 7.51

4.4 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 P{X=i},i = 1, 2, 3, ... , 8, 9, 10.

For P(X=1) : The total number of ways of ranking 10 different scores is 10!. Number of ways a female can be ranked 1 (X=1) is (#ways of choosing any one out of 5 = 5)*(#ways of arranging the other 9 = 9!) So P(X=1) = (5 x 9!)/10! = 5/10 = 1/2 For P(X=2) : The total #ways female can be ranked 2 (X=2) or less: (#ways 1st male)(#ways 1female/5)(#ways rest 8 are ranked) So P(X=2) = (5 x 5 x 8!)/10! = 5/18 and so on... P(X) = {1/2, 5/18, 5/36, 5/84, 5/252, 1/252, 0, 0, 0, 0}

4.13 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 proba- bility mass function of X, the total dollar value of all sales.

Given P(sale∣first)=0.3 P(sale∣second)=0.6 P(deluxe∣sale)=0.5 P(standard∣sale)=0.5 X is the total dollar value of all sales, which is either 0, 500, 1000, 1500, or 2000 (because the two sales are worth either 0, 500 or 1000). P(X=0)=P(no sale∣first)×P(no sale∣second)=(1−0.3)×(1−0.6)=0.7×0.4=0.28

4.46 A student is getting ready to take an important oral examination and is concerned about the possibility of hav- ing an "on" day or an "off" day. He figures that if he has an on day, then each of his examiners will pass him, inde- pendently 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 exam- iners or with 5 examiners?

He should choose 5 examiners

4.12 In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fin- gers 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. (b) Suppose that each player acts independently of the other. If each player decides to hold up the same num- ber of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1 or 2 fin- gers, what are the possible values of X and their associated probabilities?

In this task, we need to find the value that X can take and the probabilities P{X=i}. We are given the game called Two-Finger Morra, where two players show one or two fingers. If one of the players guesses the number of fingers that his opponent will show, then he wins the amount of dollars equal to the sum of the shown fingers of he and the opponent. If no one guesses correctly or both guess the opponent's number of fingers, then there is no reward. X is the amount of money won by a specific player in a single game and we are given that each player guesses the number independently, where each player shows the same number of fingers that he guesses and the probability for 1 or 2 fingers is the same.

4.47 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?

P (correct) = .87 P(convicted)= .52

4.10 Let X be the winnings of a gambler. Let p(i) =P(X = i) and suppose that p(0) = 1/3; p(1) = p(−1) = 13/55; p(2) = p(−2) = 1/11; p(3) = p(−3) = 1/165 Compute the conditional probability that the gambler wins i, i = 1, 2, 3, given that he wins a positive amount.

P(X=1 I Y) = 39/55 P(X=2 I Y) = 3/11 P(X=3 I Y) = 1/55

4.40 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?

P(X=2) = 3/8

What is a binomial random variable?

The binomial random variable may be defined as the number of "successes" in a given number of "trials" where the outcome can either be "success" or "failure." The probability of success is constant for each trial, and the trials are independent. A trial is like a mini-experiment and the final outcome is the number of successes in the series of n trials.

4.3 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.

The highest value x can take on is 6+6+6 Note, we can use symmetry to find P(X= 11...18)

pmf of a binomial rv

parameters are n and p

4.12 In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fin- gers 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 oppo- nent will hold up in such a way that each of the 4 possibili- ties is equally likely, what are the possible values of X and what are their associated probabilities?

The random variable X can take values 0 - if both players guessed each other's numbers or missed both; 2 - if you guessed correctly and both players showed one finger; 3 - if you guessed correctly and one player showed one finger and the other showed two fingers; 4 - if you guessed correctly and both players showed 2 fingers, but, also, you can get −2,−3,−4 if you guessed incorrectly, but your opponent guessed correctly, so X ∈ {−4,−3,−2,0,2,3,4}. The probability of event X=i is P(X=i) = (total number of outcomes in which X=i​)/ (total number of outcomes) P(X=+/-4) = 1/16 P(X=+-3) = 2/16 = 1/8 P(X=+-2) = 1/16 P(X=0)= 8/16 = 1/2

variance of a random variable X,

The variance, which is equal to the expected square of the difference between X and its expected value, is a measure of the spread of the possible values of X. Another identity var(X) = E[X^2] - [E(X)]^2

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

X has negative binomial distribution

4.7 Suppose that a die is rolled twice. What are the possible values that the following random variables can take on (c) the sum of the two rolls; Also calculate the probability

X ∈ {2,3,4,5,6,7,8,9,10,11,12}

4.7 Suppose that a die is rolled twice. What are the possible values that the following random variables can take on (d) the value of the first roll minus the value of the second roll? Also calculate the probability

X∈{−5,−4,−3,−2,−1,0,1,2,3,4,5}

4.30 A person tosses a fair coin until a tail appears for the first time. If the tail appears on the nth flip, the person wins 2^n dollars. Let X denote the player's winnings. Show that E[X] = +-infinity. 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?

a) No b) yes

4.7 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 Also calculate probability

a) each of the 6 numbers from the die can be a maximum number for either, so, there are 6 possibilities for X∈{1,2,3,4,5,6} b) same as part a X∈{1,2,3,4,5,6}

4.58 A certain typing agency employs 2 typists. The aver- age 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.

about .03

4.63 The number of times that a person contracts a cold in a given year is a Poisson random variable with param- eter λ = 5. Suppose that a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ = 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 individ- ual 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?

about .89

Expected value of discrete rv, X

aka mean or expectation of X

4.20 A gambling book recommends the following "win- ning strategy" for the game of roulette: Bet $1 on red. If red appears (which has probability 18/38), then take the $1 profit and quit. If red does not appear and you lose this bet (which has probability 20/38 of occurring) make addi- tional $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. (b) Are you convinced that the strategy is indeed a "win- ning" strategy? Explain your answer! (c) Find E[X].

b) No, the strategy is not the winning one. Even though that the probability that the gambler wins in this system is greater that 1/2, we will show in (c) that the expected winnings is less than zero since there is a big risk of losing all three bucks. c) E[X] = -.11