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226-f2013midterm3
8.For atrip youpack 5shirts and 3pairs ofpants, and two jackets.
Assumingeverything canbewornwitheverything else,howmanydif-ferentcombinations canyouwear?
9.Supposeyouhave6pictures, andyouwanttoarrange4ofthemalongawall.Inhowmanydifferentwayscanyouarrange them?
10.Abasketball squadhas12players.
(a) Ifallplayerscanplayanyposition,howmanywayscanateamoffivebechosen?
(b)
Ifonlyfourplayersoutofthe12areabletobethecenter, howmanywayscanateam bechosen?
11.Aboxcontains 3marbles: 1red, 1green, and 1blue. Consider anexperiment thatconsists oftaking 1marble fromthe boxand thenreplacingitintheboxanddrawingasecondmarblefromthebox.
(a) Describethesamplespace.
(b) Describethesamplespace,whenthesecondmarbleisdrawnwith-outreplacingthefirstmarble.
12.Agameconsisting offlippingacoinend
swhenthe player getstwo headsinarow,twotailsinarow,orflipsthecoinfourtimes.
(a) Drawatreediagramtoshowthewaysinwhichthegamecanend.(b) Inhowmanywayscanthegameend?
13.Twodicearethrown. LetEbethe eventthat the sumofthe diceisodd,letFbetheeventthat atleastoneofthedicelandson1,
andletGbetheeventthat thesumis5.Describetheevents
(a) E
∩F (b) E∪F(c) F
∩G(d) E
∩F0(e)E
∩F∩G 14.Threecoinsaretossed.
(a) Listtheelementsinthesamplespace.
(b) Findtheprobability that exactlytwoheadsshow.
15.Youflipanunfaircoin,
4 4
wherep(heads) =
3 andp(tails)=
1 ,tentimes.
(a) Findp(exactly 9heads). (b) Findp(exactly 7heads). (c) Findp(at least7heads).
16.Atotal of500married workingcoupleswerepolledabouttheirannualsalaries,withthefollowinginformation resulting:
lessthen $25,000morethen $25,000lessthen $25,000212198morethen $25,0003654
Wife
Husb
end Forinstance,in36ofthecouples,thewifeearnedmoreandthehusbandearnedlessthan$25,000.Ifoneofthecouples israndomlychosen,whatis
(a) theprobability that thehusband earnslessthan $25,000?
(b) theconditionalprobabilitythat thewifeearnsmorethan $25,000giventhatthehusband earnsmorethan thisamount?
(c)theconditionalprobability that thewifeearnsmorethan $25,000giventhatthehusband earnslessthan thisamount?
17.Iftwofairdicearerolled,what istheconditional probability that thefirstonelandson6giventhat thesumofthediceis9?
18.Whatisthe probability that afaircoinlands Heads4times out of5flips?
3
19.Ataparty,
1
oftheguestarewomen.75%percentofthewomenworesandalsand40%ofthemenworesandals.
(a) Whatisthe probability that apersonchosenat random at theparty isamanwearingsandals?
(b) Whatistheprobability that apersonchosenatrandomiswearingsandals?
20.Amanufactureroffrontlightsforautomobilestestslampsunderahigh-humidity,high-temperatureenvironmentusingintensity and
usefullife
astheresponsesofinterest. Thefollowingtableshowstheperformanceof140lamps:
usefullife
|
satisfactory |
unsatisfactory |
|
satisfactory intensity |
|
|
unsatisfactoryintensity |
(a) Findtheprobability that arandomly selectedlampwillyieldsat-isfactoryusefullifeandsatisfactory intensity.
(b) Find theprobability that arandomly selectedlamphassatisfac-toryusefullifegiventhat ithassatisfactory intensity.
NAME:
MAT 226: Review Test III
1. A small community consists of
10
women, each of whom has 3 children.
If one woman and one of her children are to be chosen as mother and child of the year, how many different choices are possible?
2. A college planning committee consists of 3 freshmen, 4 sophomores,
5
juniors, and 2 seniors. A subcommittee of 4, consisting of 1 person from each class, is to be chosen. How many different subcommittees are possible?
3. How many different batting orders are possible for a baseball team consisting of 9 players?
4. A class in probability theory consists of 6 men and 4 women.
An examination is given, and the students are ranked according to their performance. Assume that no two students obtain the same score.
(a) How many different rankings are possible?
(b) If the men are ranked just among themselves and the women just among themselves, how many different rankings are possible?
5. A committee of 3 is to be formed from a group of
20
people. How many different committees are possible?
6. From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed?
What if 2 of the men are feuding and refuse to serve on the committee together?
7. Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many different arrangements are possible?
8. For a trip you pack 5 shirts and 3 pairs of pants, and two jackets.
Assuming everything can be worn with everything else, how many dif- ferent combinations can you wear?
9. Suppose you have 6 pictures, and you want to arrange 4 of them along a wall. In how many different ways can you arrange them?
10. A basketball squad has 12 players.
(a) If all players can play any position, how many ways can a team of five be chosen?
(b) If only four players out of the 12 are able to be the center, how many ways can a team be chosen?
11. 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.
(a) Describe the sample space.
(b) Describe the sample space, when the second marble is drawn with- out replacing the first marble.
12. A game consisting of flipping a coin ends when the player gets two heads in a row, two tails in a row, or flips the coin four times.
(a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end?
13. 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
(a) E ∩ F (b) E ∪ F (c) F ∩ G (d) E ∩ F 0
(e) E ∩ F ∩ G
14
. Three coins are tossed.
(a) List the elements in the sample space.
(b) Find the probability that exactly two heads show.
15
. You flip an unfair coin,
where p(heads) =
3
and p(tails) = 1 , ten times.
(a) Find p(exactly 9 heads). (b) Find p(exactly 7 heads). (c) Find p(at least 7 heads).
16. A total of
50
0 married working couples were polled about their annual salaries, with the following information resulting:
Wife
Husbend
For instance, in 36 of the couples, the wife earned more and the husband earned less than $25, 000. If one of the couples is randomly chosen, what is
(a) the probability that the husband earns less than $25, 000?
(b) the conditional probability that the wife earns more than $25, 000 given that the husband earns more than this amount?
(c) the conditional probability that the wife earns more than $25, 000 given that the husband earns less than this amount?
17. 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 9?
18
. What is the probability that a fair coin lands Heads 4 times out of 5 flips?
19. At a party, 1 of the guest are women. 75% percent of the women wore sandals and 40% of the men wore sandals.
(a) What is the probability that a person chosen at random at the party is a man wearing sandals?
(b) What is the probability that a person chosen at random is wearing sandals?
20. A manufacturer of front lights for automobiles tests lamps under a high- humidity, high-temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 140 lamps:
u seful life
satisfactory
unsatisfactory
s atisfactory intensity
76
1
4
u nsatisfactory intensity
3
2
(a) Find the probability that a randomly selected lamp will yield sat- isfactory useful life and satisfactory intensity.
(b) Find the probability that a randomly selected lamp has satisfac- tory useful life given that it has satisfactory intensity.
21. An exam consists of ten true-or-false questions. If a student guesses at every answer, what is the probability that he or she will answer exactly six questions correctly?
22. A survey is done of people making purchases at a gas station
|
buy drink |
no drink |
Total |
|||
|
buy gas |
20 | 15 |
35 |
||
|
no gas |
10 | 5 | |||
|
30 |
50 |
(a) What is the probability that a person buys a drink?
(b) What is the probability that a person doesnt buy a drink? (c) What is the probability that a person buys gas and a drink?
(d) What is the probability that a person buys gas but not a drink?
(e) What is the probability that a person who buys a drink also buys gas?
(f ) What is the probability that a person who doesnt buy a drink buys gas?
23. Two fair dice are thrown. Let E denote the event that the sum of the dice is 7. Let F denote the event that the first die equals 4. Are E and F are independent events?
24. Celine is undecided as to whether to take a French course or a chem- istry course. She estimates that her probability of receiving an A grade
would be 1
in a French course and 2
in a chemistry course. If Ce-
line decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry?
25. A medical experiment showed the probability that a new medicine was effective was .75, the probability of a certain side effect was .4, and the probability of both occurring was .3. Are the events independent?
26. In Orange County, 51% of the adults are males. (It doesn’t take too much advanced mathematics to deduce that the other 49% are females.) One adult is randomly selected for a survey involving credit card usage.
(a) Find the prior probability that the selected person is a male.
(b) It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of fe- males smoke cigars. Use this additional information to find the probability that the selected subject is a male.
27. An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The companys statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability
0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. If we assume that 30% of the population is accident prone.
(a) What is the probability that a new policyholder will have an ac- cident within a year of purchasing a policy?
(b) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone?
28. (6 pt) A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. What the company does not tell you is that 15% of all steroid- free individuals also test positive (the false positive rate). 10% of the rugby team members use steroids.
Your friend on the rugby team has just tested positive. What is the probability that your friend use steroid?
29. (5 pt each) X is the outcome when we roll a pair fair dice. (a) Find E[X ] the expected value.
(b) Find V (X ) the variance of X .
30. A pair of dice, each with the numbers 1,2,2,3,3,3 on its six sides are rolled.
(a) What is the expected value of the sum of the numbers showing? (b) What is the V (X ) the variance ?
31. A contestant on a quiz show is presented with two questions, questions
1 and 2, which he is to attempt to answer in some order he chooses. If he decides to try question i first, then he will be allowed to go on to question j, j = i, only if his answer to question i is correct. If his initial answer is incorrect, he is not allowed to answer the other question. if he is 60% certain of answering question 1, worth$200, correctly and he is 80% certain of answering question 2, worth $100, correctly. Which question he should try first as to maximize his expected winnings?
32
. Let X denote a random variable that takes on any of the values ?1, 0, and 1 with respective probabilities
P (X = −1) = 0.2, P (X = 0) = 0.5, P (X = 1) = 0.3
Compute E(X 2 )
33. In a gambling game, a woman is paid $3 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should she pay to play if the game is fair?
34. Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2, or 3 power failures will strike a certain subdivision in any given year.
Find the mean and variance of the random variable X representing the number of power failures striking this subdivision.
35. The probability that a certain kind of component will survive a shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive.
The probability that a patient recovers from a rare blood disease is
0.4. If 15 people are known to have contracted this disease, what is the probability that
(a) at least 10 survive? (b) from 3 to 8 survive? (c) exactly 5 survive?
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