141 homework problems, 12B-copyright Joe Kahlig Chapter 8, Page 1Chapter 8 Homework ProblemsCompiled by Joe KahligSection 8.11. Classify the random variable as finite discrete, infinitediscrete, or continuous.(a) X = the number of games that I won in my lasttennis match.(b) X = the number of times you can buy lottery ticketsand not win the grand prize.(c) X = The amount of time spent waiting in line tobuy tickets to a concert.(d) X = The number of good jokes that I tell in asemester.(e) X = The temperature on a summer day.2. Classify the random variable as finite discrete, infinitediscrete, or co ntinuous. Also give all valid values of therandom variable.(a) X = the number of cards drawn from a well-shuffleddeck of 52 cards and cards are drawn without re-placement until a hear t is drawn.(b) X = the amount of time spent studying yesterdayfor the exam the day be fore the test.(c) X = the number of rolls it takes to get a five on asix sided die.3. The histogram for the random variable X is only missingone rectangle at X = 3probability0.050.10.150.20.253 421 560.30.35XCompute P (X ≥ 3) =4. The histogram for the random variable X is only missingtwo rectangles at X = 3 and at X = 6. We also knowthat the result X = 3 is twice as likely to happen as theresult X = 6probability0.050.10.150.20.253 421 560.30.35XCompute P (X ≥ 4) =5. The number of students waiting for help at their profes-sor’s office was recorded at various times during the day.The following data was recordedstudents 0 1 2 4 6frequency 4 10 5 4 2(a) Find the probability distribution for the randomvariable X where X is the number of students wait-ing for help.(b) Draw the histogram representing the probabilitydistribution.6. Each word of the slogan “Get ahead, learn finite math.It’s fun.” is written on a card and is dr awn a t random.The random variable X is the number of letters in theword drawn.(a) Find the probability distributio n of X.(b) Draw a histogra m for X.7. A city police department radar unit recorded the follow-ing speeds one afternoon on 17th street:30 46 53 28 52 39 34 2942 27 48 33 37 29 44 4238 47 31 51 40 31 36 4941 26 50 39 35 30 45 4338 41 36 28 52 34 37 43(a) Make a frequency table re presenting this data. Usecategory intervals of 5 miles per hour.(b) Make a probability distribution table using your an-swer in part a).8. An exam is given and the following grades.90 89 79 68 59 48 39 8489 79 67 57 42 30 99 7489 79 67 57 41 30 89 6379 66 54 40 98 88 79 9666 53 97 87 78 66 51 8397 85 77 64 50 97 84 7275 63 97 84 75 63 96 94(a) Make a frequency table re presenting this data. Usethe typical category intervals for grades: i.e. 9 0’s,80’s,....(b) Make a probability distribution table using your an-swer in part a).9. A numbe r is selected at random from the numb e rs 1, 2,3, 4, 5, 6 , 7, and 8. The random variable X is the re -mainder when the selected number is divided by 3. Findthe probability distribution of X.141 homework problems, 12B-copyright Joe Kahlig Chapter 8, Page 210. An unfair coin, P (H) =13, is toss e d until a head appearsor until it has been toss e d three times. Let the num-ber of tosses be the random variable X. Determine theprobability distribution of X.11. Three cards are drawn from a deck of car ds without re-placement. Let X be the numbe r of aces dr awn. Com-pute.(a) P (X = 0)(b) P (X = 2)12. A box has 3 yellow, 5 gray, and 4 black balls. If threeballs ar e drawn at the same time (i.e. no t replaced af-ter drawn). Let X = the number of gray balls drawn.Compute the following.(a) P (X = 2)(b) P (X ≤ 2)Section 8.213. Use the probability distribution to answer the following.X 1 2 4 5P(X) 0.3 0.15 0.35 0.2(a) What is the expected va lue of X?(b) Draw a histogra m for the probability distribution.14. Find the expected value for the random var iable XX 30 32 46 49 63 70prob 0.31 0.25 0.29 0.06 0.0415. Ten cards of a children’s game are numb ered with allpossible pairs of two different numbers from the set{1, 2, 3, 4, 5} A child draws a card. The ra ndom var i-able is the score of the card drawn. The score is 10 if thetwo numbers on the card add to 5; otherwise, the scoreis the smaller number on the car d.(a) Find the pr obability distribution of the randomvariable.(b) Compute the expected value of the random variable.16. Two cards are drawn from a deck of cards without re-placement. The random variable X is the number ofhearts drawn.(a) Find the probability distribution of X.(b) Find the expected number of hearts drawn.17. An experiment consists of rolling two fair six sideddie(one red and one green) and recording the large num-ber that is rolled. Let the random variable X be thenumber that is recorded.(a) Find the probability distribution of X.(b) Find the exp ected value of this exp e riment.18. In a lottery, 500 tickets are sold fo r $1 e ach. One firstprize of $ 2000, 1 second prize of $500, 3 third prizes of$100, and 10 co nsolation prizes of $25 are to be awarded.Let the random variable X = the net winnings o f theperson who buys only one ticket.(a) Find the probability distributio n of X.(b) Compute the expected net winnings.19. A plant manufactures microchips, 5% of which are de-fective. The plant makes a profit of $18 on each goodmicrochip and loses $23 on each defective one. What isthe expected profit on a microchip?20. Suppose you pay $5 to play a game where you toss 3fair coins. You receive $1 if one he ad results, $4 if twoheads result, and $9 if three heads result. What are yourexpected net winnings?21. A ga me cost $1.50 to play. T his game consists of anunfair coin (P (H) =37) and a fair 4 sided die. To playthe game you start off by flipping the coin. If the coincomes up tails, the game is over. If it comes up heads,you roll the die to see how many do llars that you willwin. If X is the net winnings of the person playing thegame,(a) Find the probability distributio n of X.(b) Find the exp ected value of X. Is the game fair?22. Here is a game where you draw two balls, without re-placement, from a box containing 4 r e d, 2 purple, and3 green. If you get only one red ball then you win $4.If you g et two red balls, then you win three times theamount that you paid to play the game (i.e. if the gamecost $6 then you would win $18). For any other result,you lose. What should be charged to make this game fair(or as fair as possible).23. You pay $5 to play a game where you will roll a die, withpayoffs as follows:roll 6 5 …
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