MT221 Final study guide(2024)

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Jun 12, 2024

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MT221 Final study guide You have to study for the final: MT221 Midterm study guide + Test # 3 + Final study guide(this study guide) Name_________________________ 29) A. If the probability that event A occurs during an experiment is 0.7, what is the probability that event a) Does not occur during that experiment? b) If the results of a probability experiment can be any integer from 16 to 28 and the probability that the integer is less than 20 is 0.78, what is the probability that the integer will be 20 or more? 30) If P (A) = 0.4, P (B) =0.5, and P (A and B) = 0.1, find P (A or B) 31) If P (A) =0.5, P (B) =0.3, and P (A and B) =0.2, find P (A or B) 32) IF P (A) =0.4, P (B) = 0.5, and P (A or B) =0.7, find P (A and B) 33) If P (A) = 0.4, P (A or B) = 0.9, and P (A and B) =0.1, FIND P (B) 34) A and B are events defined on a sample space, with P (A) =0.7 and P (B/A) = 0.4. Find P (A and B) 35) A and B are events defined on a sample space, with P (A/B) = 0.5 and P (B) =0.8 Find P (A and B) 36) A and B are events defined on a sample space, with P (A) =0.6 AND P (A and B) =0.3 Find P (B/A) 37) A and B are events defined on a sample space, with P (B)=0.5 and P(A and B) = 0.4 Find(A/B) 38) Juan lives in a large city and commutes to work daily by subway or by taxi. He takes the subway 80% of the time because it costs less, and he takes a taxi the other 20% of the time. When taking the subway, he arrives at work on time 70% of the time, whereas he makes it on time 90% of the time when traveling by taxi. A. What is the probability that Juan took the subway and is at work on time on any given day? B. What is the probability that Juan took a taxi and is at work on time on any given day? 39) Suppose that A and B are events defined on a common sample space and that the following probabilities are known: P (A) =0.3, P (B) =0.4 and P (A/B) =0.2 Find P (A or B) 40) Suppose that A and B are events defined on a common sample space and that the following probabilities are known: P (A or B) =0.7, P (B) =0.5, and P (A/B) =0.2. Find P (A) 41) Suppose that A and B are events defined on a common sample space and that the following probabilities are known: P (A)=0.4, P(B)=0.3 and P(A or B) =0.66. Find P (A/B) 42) Given P (A or B) =1.0, P ( A ∧ B ) =0.7¸ a) P (B) b). P (A) c). P (A/B) 43) Determine whether each of the following sets of events is mutually exclusive. A. Five coins are tossed: “no more than one head is observed,” “two heads are observed,” and “three or more heads are observed.” B. A salesperson calls on a client and makes a sake: the amount of the sale is “less than $100.”is between $100 and $1,000,” is “more than $500” C. One student is selected at random from the student body” the person selected is “female,” is “male,” is “older than 21,” 44) Do people take indoor swimming lessons in the middle of the hot summer? They sure do at the Webster Aquatic Center. During the month of July 2009 alone, 283 people participated in various forms of lessons. Swim Categories DaytimeEvenings Preschool 66 80 Levels 69 56 Adult and diving 10 2 Total 145 138 If one swimmer was selected at random from the July participants:

a. Are the events the selected participants is “ daytime” and “evening” mutually exclusive? Explain b. Are the events the selected participant is “preschool” and “levels” mutually exclusive? Explain c. Are the events the selected participant is “daytime” and “preschool” mutually exclusive? Explain d. Find P (preschool). e. Find P (daytime) f. Find P ( not levels) g. Find P ( preschool or evening) h. Find P ( preschool and daytime) i. Find P ( daytime/ levels) j. Find P (adult and diving/ evening) 45) Determine whether each of the following pairs of events is independent: a. Rolling a pair of dice and observing a “1” on the first die and a “1” on the second die b. Drawing a “spade” from a regular deck of playing cards and then drawing another “spade” from the same deck without replacing the first card c. Same as part b except the first card is returned to the deck without replacing the first card d. Owning a red automobile and having blonde hair e. Owning a red automobile and having a flat tire today f. Studying for an exam and passing the exam 46) Suppose that P (A) = 0.3, P (B)=0.4, and P (A and B) =0.20 a. What is P (A/B) ? b. What is P (B/A)? c. Are A and B independent? 47) One student is selected at random from a group of 200 students known to consist of 140 full-time (80 female and 60 male) students and 60 part-time (40 female and 20 male) students. Event A is “the student selected is full time,” and event C is “the student selected is female.” a. Are events A and C independent? Justify your answer. b. Find the probability P( A and C) 48) You have applied for two scholarships: a merit scholarship (M) and an athletic scholarship (A). Assume the probability that you receive the athletic scholarship is 0.25, the probability that you receive both scholarships is 0.15, and the probability that you get at least one of the scholarship is 0.37. Use a Venn diagram to answer these questions: A. What is the probability that you receive the merit scholarship? B. What is the probability that you do not receive either of the two scholarships? C. What is the probability that you receive the merit scholarship given that you have been awarded the merit scholarship? D. What is the probability that you receive the athletic scholarship, given that you have been awarded the merit scholarship? E. Are the events “receiving an athletic scholarship” and “receiving a merit scholarship” independent events? Explain. 49 Consider the set of integers 1,2,3,4, and 5. A. One integer is selected at random. What is the probability that it is odd? B. Two integers are selected at random (one at a time with replacement so that each of the five is available for a second selection). Find the probability that neither is odd; exactly one of them is odd; both are odd. 50) A box contains 25 parts, of which 3 are defective and 22 are nondefective. If 2 parts are selected without replacement, find the following probabilities: A P(both are defective) B. P (exactly one is defective) C. P (neither is defective) 51) a. Express ( x ) = 16, for x= 1, 2, 3, 4, 5, 6, in distribution form. B. Construct a histogram of the probability distribution P( x )= 16, for x= 1, 2, 3, 4, 5, 6. c. Describe the shape of the histogram in part b 52) a. Explain how the various values of x in a probability distribution form a set of mutually exclusive events b. Explain how the various values of x in a probability distribution form a set of “ all-inclusive” events. 53) Test the following function to determine whether it is a probability function. If it is nit, try to make it into a probability function. S(x) =6-x-736, for x= 2, 3, 4, 5, 6, 7...11, 12 a. List the distribution of probabilities and sketch a histogram.

b. Do you recognize S(x) ? If so, identify it. 54) a) Form the probability distribution table for P(x) = x6, for x =1, 2, 3 b. Find the extensions xP(x) and x^2 P(x) for each x. c. Find [ x^2P(x )]. d.Find the mean for P(x) =x6, for x =1, 2, 3. 55) Given the probability function P (x) =5-x10, for x =1, 2, 3, 4, find the mean and standard deviation. 56) Given the probability function R (x)= 0.2, for x =0, 1, 2, 3, 4, find the mean and standard deviation. 57) The College Board website provides much information for students, parents, and professionals with respect to the many aspects involved in Advanced Placement (AP) courses and exams. One particular annual report provides the percent of students who obtain each of the possible AP grades (1 through 5). The 2008 grade distribution for all subjects was as follows: AP Grades Percents 1 20.9 2 21.3 3 24.1 4 19.4 5 14.3 a. Express this distribution as a discrete probability distribution. b. Find the mean and standard deviation of the distribution of the AP exam scores for 2008. 58) The random variable A has the following probability distribution: A 1 2 3 4 5 P(A) 0.6 0.1 0.1 0.1 0.1 a. Find the mean and standard deviation of A . b. How much of the probability distribution is within 2 standard deviations of the mean? c. What’s the probability that A is between -2 and +2 59) The random variable x has the following probability distribution: x 1 2 3 4 5 P(x) 0.6 0.1 0.1 0.1 0.1 a. Find the mean and standard deviation of (x). b. What’s the probability that x is between - and + 60) Evaluate each of the following. a) 4! b. 7! c. 0! d.6!/2! e). 5!/2!3! f) 6 ! 4 ! ( 6 − 4 ) ! g) ( 7 3 ) h) ( 4 1 ) ¿ 61) Use the probability function for three coin tosses as demonstrated on page 239 and verify the probabilities doe x =0,2, and 3. 62) If x is a binomial random variable, calculate the probability of x for each case. A. n= 4, x=1, p=0.3 B. n= 3, x=2, p=0.8 C. n=2, x=0, p = 1 4 D. n=12, x=12, p=0.99 E. n=4, x=2, p=0.5 F. n=3, x=3, p= 1 6 63) if x is a binomial random variable, use Table 2 in Appendix B to determine the probability of x for each of the following: A. n=10, x=8, p=0.3 B. n=8, x=7, p=0.95 C. n=15, x=3, p=0.05 D. n=12, x=12, p=0.99 E. n=9, x=0, p=0.5 F. n=6, x=1, p=0.01 G. Explain the meaning of the symbol 0+ that appears in Table 2. 64) Consider the manager of Steve’s Food Markets illustrated in Example 5.9. What would be the manager’s “risk” if he bought “better” eggs, say with P(BAD) = 0.01 using the “more than one” guarantee? 65) If boys and girls are equally likely to be born, what is the probability that in a randomly selected family of six children, there will be at least one boy? (Find the answer using a formula.) 66) Consider the binomial distribution where n=11 and p-0.05. A. Find the mean and standard deviation using formulas (5.7) and (5.8). B. Using Table 2 in Appendix B, list the probability distribution and draw a histogram C. Locate µ and σ on the histogram. 67) Given the binomial probability function P (X) = 5 X ∙ ¿ ∙ ¿ 1 2 ¿¿ 5 − x for x= 0, 1, 2,3,4,5 Ch oco lat e No cho col ate Jell ybe ans Cr ea m - fil le d Mars hmall ow M al te d D o n ’t K n o w 30 % 25 % 13 % 1 1 % 8% 7 % 6 %

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A. Calculate the mean and standard deviation of the random variable by using formulas (5.1), (5.3a), and (5.4.) B. Calculate the mean and standard deviation using formulas (5.7) and (5.8) 68) If the binomial ( q + p) is squared, the results is ( q + p ) 2 = q 2 + 2qp + p 2 . For the binomial experiment with n=2, the probability of no successes in two trials is q 2 (the first term in the expansion), the probability of one success in two trails is 2qp (the second term in the expansion). And the probability of two successes in two trials is p 2 (the third term in the expansion). Find ¿ and compare its terms to the binomial probabilities for n+3 trails. 69) A binomial random variable has a mean equal to 200 and a standard deviation of 10. Find the values of n and p. 70) The probability of success on a single trial of a binomial experiment is known to be 1 4 . The random variable x, the number of successes, has a mean value of 80. Find the number of trials involved in this experiment and the standard deviation of x.

MT221 Final study guide(2024)

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