HW 5.1-5.5
If E and F are disjoint events, then P(E or F)=
P(E)+P(F)
If E and F are not disjoint events, then P(E or F)=________.
P(E)+P(F)-P(E and F)
How many different 10-letter words (real or imaginary) can be formed from the following letters? WRELPOCTKV
10!/2! = 10x9x8x7x6x5x4x3 = 1814400
A(n) ____________ is an ordered arrangement of r objects chosen from n distinct objects without repetition.
permutation
A baseball player hit 65 home runs in a season. Of the 65 home runs, 21 went to right field, 21 went to right center field, 11 went to centerfield, 10 went to left center field, and 2 went to left field. (a) What is the probability that a randomly selected home run was hit to right field? (b) What is the probability that a randomly selected home run was hit to left field? (c) Was it unusual for this player to hit a home run to left field? Explain.
(a) 21/65=0.323 (b) 0.031 (c) Yes, because left field is less than <0.05
A cheese can be classified as either raw-milk or pasteurized. Suppose that 90% of cheeses are classified as pasteurized. (a) Two cheeses are chosen at random. What is the probability that both cheeses are pasteurized? (b) Five cheeses are chosen at random. What is the probability that all five cheeses are pasteurized? (c) What is the probability that at least one of five randomly selected cheeses is raw-milk? Would it be unusual that at least one of five randomly selected cheeses is raw-milk?
(a) 0.90(0.90)=0.81 (b) 0.90^5=0.5905 (c) 1-0.5905=0.4095 It would also not be unusual.
The probability that a randomly selected 2-year-old male salamander will live to be 3 years old is 0.96523. (a) What is the probability that two randomly selected 2-year-old male salamanders will live to be 3 years old? (b) What is the probability that seven randomly selected 2-year-old male salamanders will live to be 3 years old? (c) What is the probability that at least one of seven randomly selected 2-year-old male salamanders will not live to be 3 years old? Would it be unusual if at least one of seven randomly selected 2-year-old male salamanders did not live to be 3 years old?
(a) 0.93167 (b) 0.78058 (c) 0.21942 (d)No because the probability of this happening is greater that 0.05
The probability that a randomly selected 5-year-old male chipmunk will live to be 6 years old is 0.95773. (a) What is the probability that two randomly selected 5-year-old male chipmunks will live to be 6 years old? (b) What is the probability that seven randomly selected 5-year-old male chipmunks will live to be 6 years old? (c) What is the probability that at least one of seven randomly selected 5-year-old male chipmunks will not live to be 6 years old? Would it be unusual if at least one of seven randomly selected 5-year-old male chipmunks did not live to be 6 years old?
(a) 0.95773*=0.91725 (b) 0.95773^7=0.73910 (c) 1-0.73910=0.2609 (d) No ..........Greater than 0.05
A test to determine whether a certain antibody is present is 99.8% effective. This means that the test will accurately come back negative if the antibody is not present (in the test subject) 99.8% of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is 0.002. Suppose the test is given to four randomly selected people who do not have the antibody. (a) What is the probability that the test comes back negative for all four people? (b) What is the probability that the test comes back positive for at least one of the four people?
(a) 0.992 (b) 0.008
A bag of 35 tulip bulbs contains 14 red tulip bulbs, 12 yellow tulip bulbs, and 9 purple tulip bulbs. Suppose two tulip bulbs are randomly selected without replacement from the bag. (a) What is the probability that the two randomly selected tulip bulbs are both red? (b) What is the probability that the first bulb selected is red and the second yellow? (c) What is the probability that the first bulb selected is yellow and the second red? (d) What is the probability that one bulb is red and the other yellow?
(a) 14/35 x 13/34 = 0.153 (b) 14/35 x 12/34 = 0.141 (c) 12/35 x 14/34 = 0.141 (d) 0.141 + 0.141 = 0.282
Suppose you just purchased a digital music player and have put 9 tracks on it. After listening to them you decide that you like 2 of the songs. With the random feature on your player, each of the 9 songs is played once in random order. Find the probability that among the first two songs played (a) You like both of them. Would this be unusual? (b) You like neither of them. (c) You like exactly one of them. Redo (a)-(c) if a song can be replayed before all 7 songs are played (d) The probability that you like both songs is The probability that you like neither song is The probability that you like exactly one song is
(a) 2/9 x 1/8 =0.028 (yes unusual) (b)7/9 x 6/8 = 0.583 (c)2/9 x 7/8 + 7/9 x 2/8 = 0.389 (d) 2/9 x 2/9 = 0.049 7/9 x 7/9 = 0.605 2/9 x 7/9 + 7/9 x 2/9 = 0.346
In a recent poll, a random sample of adults in some country (18 years and older) was asked, "When you see an ad emphasizing that a product is "Made in our country," are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the following contingency table. Complete parts (a) through (c). (a) What is the probability that a randomly selected individual is 35 to 44 years of age, given the individual is more likely to buy a product emphasized as "Made in our country"? (b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in our country," given the individual is 35 to 44 years of age? (c) Are 18- to 34-year-olds more likely to buy a product emphasized as "Made in our country" than individuals in general?
(a) 309/1328=0.233 (b) 309/563=0.59 (c) No, Less likely
A certain four-cylinder combination lock has 45 numbers on it. To open it, you turn to a number on the first cylinder, then to a second number on the second cylinder, and then to a third number on the third cylinder and so on until a four-number lock combination has been effected. Repetitions are allowed, and any of the 45 numbers can be used at each step to form the combination. (a) How many different lock combinations are there? (b) What is the probability of guessing a lock combination on the first try?
(a) 45^4 4100625 (b) 1/4100625
The grade appeal process at a university requires that a jury be structured by selecting seven individuals randomly from a pool of eight students and seven faculty. (a) What is the probability of selecting a jury of all students? (b) What is the probability of selecting a jury of all faculty? (c) What is the probability of selecting a jury of four students and three faculty?
(a) 8C7=8/6435 =0.00124 (b) 7C7=1/6435 =0.00016 (c) 0.38073
Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: A person attaining a position as a professor. F: The same person attaining a PhD. (b) E: A randomly selected person accidentally killing a spider. F: A different randomly selected person accidentally swallowing a spider. (c) E: The rapid spread of a cocoa plant disease. F: The price of chocolate.
(a) E and F are dependent because attaining a PhD can affect the probability of a person attaining a position as a professor. (b) E cannot affect F and vice versa because the people were randomly selected, so the events are independent. (c) The rapid spread of a cocoa plant disease could affect the price of chocolate, so E and F are dependent.
For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.16 probability of failure. Complete parts (a) through (c) below. (a) Would it be unusual to observe one component fail? Two components? (b) What is the probability that a parallel structure with 2 identical components will succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than 0.9998?
(a) It would not be unusual to observe one component fail, since the probability that one component fails, 0.16, is greater than 0.05. It would be unusual to observe two components fail, since the probability that two components fail, 0.0256 is less than 0.05. (b) 0.9744 (c) 5
The following data represent the number of different communication activities used by a random sample of teenagers in a given week. Complete parts (a) through (d). (a) Are the events "male" and "5+ activities" independent? (b) (b) Are the events "female" and "0 activities" independent? (c) Are the events "1−2 activities" and "5+activities" mutually exclusive? (d) Are the events "male" and "1−2 activities" mutually exclusive?
(a) No, because Upper P left parenthesis male right parenthesis and Upper P left parenthesis male| 5 plus activities right parenthesisP(male) and P(male|5+ activities) are not equal. (b) Yes, because Upper P left parenthesis female right parenthesis and Upper P left parenthesis female| 0 activities right parenthesisP(female) and P(female|0 activities) are equal. (c) Yes, because Upper P left parenthesis 1 minus 2 activities and 5 plus activities right parenthesisP(1−2 activities and 5+ activities is zero. (d) No, because Upper P left parenthesis male and 1 minus 2 activities right parenthesisP(male and 1−2 activities) is not zero.
According to a poll, about 16% of adults in a country bet on professional sports. Data indicates that 46.6% of the adult population in this country is male. Complete parts (a) through (e). (a) (a) Are the events "male" and "bet on professional sports" mutually exclusive? Explain. (b) Assuming that betting is independent of gender, compute the probability that an adult from this country selected at random is a male and bets on professional sports. (c) Using the result in part (b), compute the probability that an adult from this country selected at random is male or bets on professional sports. (d) The poll data indicated that 9.6% of adults in this country are males and bet on professional sports. What does this indicate about the assumption in part (b)? (e) How will the information in part (d) affect the probability you computed in part (c)? Select the correct choice below and fill in any answer boxes within your choice.
(a) No. A person can be both male and bet on professional sports at the same time. (b) .466x.16=0.0746 (c) .466+.16=0.626-0.0746 =0.5514 (d) The assumption was incorrect and the events are not independent. (e) 0.466+0.16-0.096=0.5300
What is the probability of an event that is impossible? Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible? (a) What is the probability of an event that is impossible? (b) Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible?
(a) ZERO (0) (b) NO
List all the permutations of three objects a, b, and c taken two at a time without repetition. What is 3P2?
(a) ab, ac, ba, bc, ca, cb (b) 6
List all the combinations of three objects x, y, and z taken two at a time. What is 3C2?
(a) xy, xz, yz, (b) 3
According to a certain country's department of education, 40.5% of 3-year-olds are enrolled in day care. What is the probability that a randomly selected 3-year-old is enrolled in day care?
0.405
Suppose that E and F are two events and that P(E and F)=0.2 and P(E)=0.4. What is P(F|E)?
0.5
If P(E)=0.45, P(E or F)=0.65, and P(E and F)=0.10, find P(F).
0.65-0.45=0.20 0.20+0.10=0.30
9C8=
9!/8! (9-8)! = 9!/8!(1)!=9
Suppose Jim is going to build a playlist that contains 14 songs. In how many ways can Jim arrange the 14 songs on the playlist?
14! = 87178291200
Four members from a 27-person committee are to be selected randomly to serve as chairperson, vice-chairperson, secretary, and treasurer. The first person selected is the chairperson; the second, the vice-chairperson; the third, the secretary; and the fourth, the treasurer. How many different leadership structures are possible?
27!/(27-4)!=27!/23!=421200
Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E={1, 3, 5, 6}.
4/10=2/5=.40
8P2 =
8!/(8-2)! = 8!/6! =8X7=56
A golf ball is selected at random from a golf bag. If the golf bag contains 6 green balls, 2 brown balls, and 13 black balls, find the probability of the following event. The golf ball is green or brown.
8/21=0.381
A woman has nine skirts and four blouses. Assuming that they all match, how many different skirt-and-blouse combinations can she wear?
9 x 4 =36
The notation P(F E) means the probability of event ________ given event _______.
F given event E
Two events E and F are ________ if the occurrence of event E in a probability experiment does not affect the probability of event F.
Independent
You suspect a 6-sided die to be loaded and conduct a probability experiment by rolling the die 400 times. The outcome of the experiment is listed in the following table. Do you think the die isloaded? Why?
No because each value has an approximately equal chance of occurring.
A probability experiment is conducted in which the sample space of the experiment is S={7,8,9,10,11,12,13,14,15,16,17,18}. Let event E={8,9,10,11,12,13} and event F={12,13,14,15}. List the outcomes in E and F. Are E and F mutuallyexclusive?
{12,13} No E and F has outcomes in common
In a certain card game, the probability that a player is dealt a particular hand is 0.43. Explain what this probability means. If you play this card game 100 times, will you be dealt this hand exactly 43 times? Why or why not?
The probability 0.43 means that approximately 43 out of every 100 dealt hands will be that particular hand. No, you will not be dealt this hand exactly 43 times since the probability refers to what is expected in the long-term, notshort-term.
Why is the following not a probability model? Color. Probability Red. 0.2 Green. −0.3 Blue. 0.1 Brown 0.3 Yellow. 0.3 Orange. 0.4
This is not a probability model because at least one probability is less than 0.
Use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to determine which of the assignments of probabilities are consistent with the definition of a probability model.
A,B,C,F is/are consistent with the definition of a probability model. A-1/4, 1/4,1/4,1/4 B-0,0,0,1 C-1/9,1/9,7/18,7/19 F-1/8,1/4,1/4,3/8