Problem 4: Teen smoking

According to a report published by the Centers for Disease Control and Prevention, about 20 percent of high school students currently use a tobacco product. This number of down from 23 percent in 2014. (https://www.cdc.gov/tobacco/data_statistics/fact_sheets/youth_data/tobacco_use/index.htm). We would like to conduct a study to evaluate high school students’ attitudes toward scenes of smoking in the movies. Suppose you randomly select students to survey them on their opinion about this question.

a) What is the probability that none of the first 4 students you interview is a smoker? (1 point)

b) What is the probability that there are no more than two smokers among 10 students you randomly choose? (2 points)

c) What is the probability that exactly 3 out of a new sample of 10 students do not smoke? (1 point)

a box contains two halves separated by a partition. Initially, there are 3 ideal gas molecules in the left half, and a vacuum in the right half. The partiition is then removed so the gas goes through a "free expansion" so that the speeds of the molecules are unaffected. In the final state, each molecule has probability of 1/2 of being in the left half of the box and probability of 1/2 being in the right half od the box. Because each of the 3 molecules now has twice as many possible positions than before, the number of microstates of the system has increased by a factor of 2x2x2=2^3. Determine the change in entropy of the system as a result of removing the partition. (Note: We can only calculate the change in entropy of the system but not the initial and final entropy values without knowing more information) What is the probability that all three molecules will simultanesouly be in the left half of the box?

The following table presents the number of reports of graffiti in each of New York's five boroughs over a one-year period. These reports were classified as being open, closed, or pending. A graffiti report is selected at random. Compute the following probabilities. Round final answer to four decimal places. Borough Open Reports Closed Reports Pending Reports Total Bronx 1121 1622 80 2823 Brooklyn 1170 2706 48 3924 Manhattan 744 3380 25 4149 Queens 1353 2043 25 3421 Staten Island 83 118 0 201 Total 4471 9869 178 14,518 Source: NYC OpenData Send data to Excel Part 1 of 6 (a) The report is closed and comes from Staten Island. The probability that the report is closed and comes from Staten Island is . Part 2 of 6 (b) The report is open or comes from Bronx. The probability that the report is open or comes from Bronx is . Part 3 of 6 (c) The report comes from Manhattan. The probability that the report comes from Manhattan is . Part 4 of 6 (d) The report that does not come from Manhattan. The probability that the report does not come from Manhattan is . Part 5 of 6 (e) The report is pending. The probability that the report is pending is . Part 6 of 6 (f) The report is from the Brooklyn or Queens. The probability that the report is from the Brooklyn or Queens is .

A recent study estimates that 45% of iPhone still have their phone their phone within 2 years of purchasing it. Suppose you randomly select 30 iPhone users. Let random variable X denote the number of iPhone users who still have their original phone after 2 years.

1)Describe the probability distribution of X (Hint: Give the name of the distribution and identify n and p).

2)Find the expected value of X. Round to 1 decimal place.

3)Find the standard deviation of X. Round to 3 decimal places.

4)Determine the probability that X equals 13. Round to 3 decimal places.

5)Determine the probability that EXACTLY 11 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

6)Determine the probability that at most 5 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

7)Determine the probability that at least 8 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

8)Determine the probability that between 6 and 9 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

*only need answers for questions 1,6,7,8. Thank you!

1. Let K be the index of the first successful trial that is immediately followed by a failure. In symbols, (3.53) K = inf{n ∈ Z>0 : Xn = 1, Xn+1 = 0}. Find the probability mass function of K. Check that your answer is a legitimate probability mass function. Hint. Decompose the event {K = m} into disjoint components expressed in terms of the trial outcome variables {Xi}. Note that a success before m cannot be immediately followed by a failure.

2. Let N be the index of the first success, as defined in (3.18). Let K be the index of the first success that is immediately followed by a failure, as defined in (3.53). (a) Find the probability P(XN+1 = 0, XN+2 = 1, XN+3 = 0). (b) Find the probability P(XK+1 = 0, XK+2 = 1, XK+3 = 0). The task is to find the probability that the three trials immediately following the random index K yield a failure, a success, and a failure, in that order. Explain why your answer makes intuitive sense.

3. Assume that 0 < p < 1. (a) Let Sn ∼ Bin(n, p) count the number of successes in the first n trials. Fix a positive integer k. Show that lim n→∞ P(Sn ≤ k) = 0. (b) Show that in infinitely many trials there are infinitely many successes with probability one.

4. (a) Susan tries to exercise at "Pure Fit" Gym each day of the week, except on the weekends.(Saturdays and Sundays). Susan is able to exercise, on average, on 75% of the weekdays.(Monday to Friday).

i. Find the expected value and the standard deviation of the number of days she

exercises in a given week.

ii. Given that Susan exercises on Monday, _nd the probability that she will exercise at

least 3 days in the rest of the week.

iii. Find the probability that in a period of four weeks, Susan exercises 3 or less days in only two of the four weeks.

(b) A car repair shop uses a particular spare part at an average rate of 6 per week. Find the probability that:

i. at least 6 are used in a particular week.

ii. exactly 18 are used in a 3-week period.

iii. exactly 6 are used in each of 3 successive weeks.

(c) The breaking strength (in pounds) of a certain new synthetic piece of glass is normally distributed, with a mean of 115 pounds and a variance of 4 pounds.

i. What is the probability that a single randomly selected piece of glass will have

breaking strength between 118 and 120 pounds?

ii. A new synthetic piece of glass is considered defective if the breaking strength is less than 113.6 pounds. What is the probability that a single randomly selected piece of 3 glass will be defective?

iii. What is the probability that out of 200 pieces of randomly selected glass, more than 55 of them are defective.

2. (a)

Joe tries to study at a study cafe each day of the week, except on the weekends (Saturdays & Sundays). Joe is able to study, on average, on 75% of the weekdays (Monday-Friday).

i. Find the expected value and the standard deviation of the number of days he studies in a given week.

ii. Given that Joe studies on Monday, find the probability that he will study at least 3 days in the rest of the week.

iii. Find the probability that in a period of four weeks, Joe studies 3 or less days in only two of the four weeks.

(b)

A bakery uses a particular ingredient at an average rate of 6 per week. Find the probability that:

i. at least 6 are used in a particular week.

ii. exactly 18 are used in a 3-week period.

iii. exactly 6 are used in each of 3 successive weeks.

(c)

The breaking strength (grams) of a certain ceramic is normally distributed, with a mean of 115 grams and a variance of 4 grams.

i. What is the probability that a single randomly selected piece of ceramic will have breaking strength between 118 and 120 grams?
ii. A new piece of ceramic is considered defective if the breaking strength is less than 113.6 grams. What is the probability that a single randomly selected piece of ceramic will be defective?

iii. What is the probability that out of 200 pieces of randomly selected ceramic, more than fifty five of them are defective.

According to Nielsen Media Research, the average number of hours of TV viewing by adults (18 and over) per week in the United States is 36.07 hours. Suppose the standard deviation is 8.7 hours and a random sample of 51 adults is taken. Appendix A Statistical Tables a. What is the probability that the sample average is more than 36 hours? b. What is the probability that the sample average is less than 36.6 hours? c. What is the probability that the sample average is less than 29 hours? If the sample average actually is less than 40 hours, what would it mean in terms of the Nielsen Media Research figures? d. Suppose the population standard deviation is unknown. If 66% of all sample means are greater than 35 hours and the population mean is still 36.07 hours, what is the value of the population standard deviation?

a. What is the probability that the sample average is more than 36 hours?
b. What is the probability that the sample average is less than 36.6 hours?
c. What is the probability that the sample average is less than 29 hours? If the sample average actually is less than 40 hours, what would it mean in terms of the Nielsen Media Research figures?
d. Suppose the population standard deviation is unknown. If 66% of all sample means are greater than 35 hours and the population mean is still 36.07 hours, what is the value of the population standard deviation?

​a) Determine the probability of selecting a female.

​(Round to three decimal places as​ needed.)

​b) Determine the probability of selecting a student that is 22 to 24 years old.

​(Round to three decimal places as​ needed.)

​c) Determine the probability of selecting a woman who is 35 years old or older.

​(Round to three decimal places as​ needed.)

​d) Determine the probability of selecting either a student who is a woman or is 25 to 29 years old.

​(Round to three decimal places as​ needed.)

​e) Determine the probability of selecting a​ man, given that the student is 22 to 24 years old.

​(Round to three decimal places as​ needed.)

​f) Determine the probability of selecting a student that is 30 to 34 years​ old, given that the student is a man.