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Probability and the bell curve exercises · Physics Playground

Probability and the bell curve

145 problems with worked solutions, from Statistics.

Start with a random one

Learn it first: Probability: Counting the Chances, The Bell Curve & Standard Deviation

Statistics3.1 Terminology

Example 3.1The sample space S is the whole numbers starting at one and less than 20. S = ________ Let event A = the even numbers and event B = numbers greater than 13. A = ________, B = ________ P(A) = ________, P(B) = ________…
Example 3.2A fair, six-sided die is rolled. The sample space, S, is {1, 2, 3, 4, 5, 6}. Describe each event and calculate its probability. Event T = the outcome is two. Event A = the outcome is an even number. Event B = the…
Example 3.3Let’s denote the events M = the subject is male, F = the subject is female, R = the subject is right-handed, L = the subject is left-handed. Compute the following probabilities: P(M) P(F) P(R) P(L) P(M AND R) P(F AND…

Statistics3.2 Independent and Mutually Exclusive Events

Example 3.5You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q…
Example 3.7Flip two fair coins. Find the probabilities of the events. Let F = the event of getting at most one tail (zero or one tail). Let G = the event of getting two faces that are the same. Let H = the event of getting a…
Example 3.8Are C and E mutually exclusive events? Answer yes or no. Why or why not?
Example 3.9 (part 1)a. Show that P(G|H) = P(G).
Example 3.9 (part 2)b. Show P(G AND H) = P(G)P(H).
Example 3.10Let event C = taking an English class. Let event D = taking a speech class. Suppose P(C) =.75, P(D) =.3, P(C|D) =.75 and P(C AND D) =.225. Justify your answers to the following questions numerically. Are C and D…
Example 3.12In a particular class, 60 percent of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, 75…
Example 3.13Toss one fair coin (the coin has two sides, H and T). The outcomes are ________. Count the outcomes. There are ________ outcomes. Toss one fair, six-sided die (the die has 1, 2, 3, 4, 5, or 6 dots on a side). The…

Statistics3.3 Two Basic Rules of Probability

Example 3.15 (part 1)a. What is the probability that he makes both goals?
Example 3.15 (part 2)b. What is the probability that Carlos makes either the first goal or the second goal?
Example 3.15 (part 3)c. Are A and B independent?
Example 3.15 (part 4)d. Are A and B mutually exclusive?
Example 3.16 (part 1)a. What is the probability that the member is a novice swimmer?
Example 3.16 (part 2)b. What is the probability that the member practices four times a week?
Example 3.16 (part 3)c. What is the probability that the member is an advanced swimmer and practices four times a week?
Example 3.16 (part 4)d. What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and being an intermediate swimmer mutually exclusive? Why or why not?
Example 3.16 (part 5)e. Are being a novice swimmer and practicing four times a week independent events? Why or why not?
Example 3.17What is the probability that Felicity enrolls in math and speech? Find P(M AND S) = P(M|S)P(S). What is the probability that Felicity enrolls in math or speech classes? Find P(M OR S) = P(M) + P(S) − P(M AND S). Are…
Example 3.18 (part 1)a. What is the probability that the woman develops the disease? What is the probability that woman tests negative?
Example 3.18 (part 2)b. Given that the woman develops the disease, what is the probability that she tests negative?
Example 3.18 (part 3)c. What is the probability that the woman has the disease AND tests negative?
Example 3.18 (part 4)d. What is the probability that the woman has the disease OR tests negative?
Example 3.18 (part 5)e. Are having the disease and testing negative independent events?
Example 3.18 (part 6)f. Are having the disease and testing negative mutually exclusive?
Example 3.19Refer to the information in Example 3.18. P = tests positive. Given that a woman develops the disease, what is the probability that she tests positive? Find P(P|B) = 1 − P(N|B). What is the probability that a woman…

Statistics3.4 Contingency Tables

Example 3.20Find P(Person uses a cell phone while driving). Find P(Person had no violation in the last year). Find P(Person had no violation in the last year and uses a cell phone while driving). Find P(Person uses a cell phone…
Example 3.21 (part 1)a. Complete the table.
Example 3.21 (part 2)b. Are the events being female and preferring the coastline independent events? Let F = being female and let C = preferring the coastline. Find P(F AND C). Find P(F)P(C). Are these two numbers the same? If they are,…
Example 3.21 (part 3)c. Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let M = being male, and let L = prefers hiking near lakes and streams. What word tells you this is a…
Example 3.21 (part 4)d. Find the probability that a person is female or prefers hiking on mountain peaks. Let F = being female, and let P = prefers mountain peaks. Find P(F). Find P(P). Find P(F AND P). Find P(F OR P).
Example 3.22 (part 1)a. Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is 1.
Example 3.22 (part 2)b. What is the probability that Alissa does not catch Muddy?
Example 3.22 (part 3)c. What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa?
Example 3.23TOTAL each column and each row. Total data = 4,520.7. Find P(2009 AND Crime A). Find P(2010 AND Crime B). Find P(2010 OR Crime B). Find P(2011|Crime A). Find P(Crime D|2008).

Statistics3.5 Tree and Venn Diagrams

Example 3.24 (part 1)a. List the 24 BR outcomes: B1R1, B1R2, B1R3,...
Example 3.24 (part 2)b. Calculate P(RR).
Example 3.24 (part 3)c. Calculate P(RB OR BR).
Example 3.24 (part 4)d. Using the tree diagram, calculate P(R on 1st draw AND B on 2nd draw).
Example 3.24 (part 5)e. Using the tree diagram, calculate P(R on 2nd draw GIVEN B on 1st draw).
Example 3.24 (part 6)f. Using the tree diagram, calculate P(BB).
Example 3.24 (part 7)g. Using the tree diagram, calculate P(B on the 2nd draw GIVEN R on the first draw).
Example 3.25 (part 1)a. P(RR) = ________
Example 3.25 (part 2)b. Fill in the blanks. P(RB OR BR) = ( 3/(11))( 8/(10)) + (________)(________) = (48)/(110)
Example 3.25 (part 3)c. Because this is a conditional probability, we restrict the sample space to consider only those outcomes that have a blue marble in the first draw. Look at the second level of the tree to see that P(R on 2nd|B on…
Example 3.25 (part 4)d. Fill in the blanks. P(R on 1st AND B on 2nd) = P(RB) = (________)(________) = (24)/(110)
Example 3.25 (part 5)e. Find P(BB).
Example 3.25 (part 6)f. Find P(B on 2nd|R on 1st).
Example 3.26Which shows the probability that both kittens are tabby? a.( 1/2)( 1/2) b.( 4/9)( 4/9) c.( 4/9)( 3/8) d.( 4/9)( 5/9) What is the probability that one kitten of each coloring is selected? a.( 4/9)( 5/9) b.( 4/9)( 5/8)…
Example 3.29Forty percent of the students at a local college belong to a club and 50 percent work part time. Five percent of the students work part time and belong to a club. Draw a Venn diagram showing the relationships. Let C…With a figure
Example 3.30A person with type O blood and a negative Rh factor (Rh–) can donate blood to any person with any blood type. Four percent of African Americans have type O blood and a negative Rh factor, 5−10 percent of African…With a figure

Statistics4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

Example 4.2 (part 1)Describe the random variable in words. Let X = the number of days Nancy ________.
Example 4.2 (part 2)b. In this example, what are possible values of X?
Example 4.2 (part 3)c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in Example 4.1. The table should have two columns labeled x and P(x).

Statistics4.2 Mean or Expected Value and Standard Deviation

Example 4.6 (part 1)a. Define a random variable X.
Example 4.6 (part 2)b. Complete the following expected value table. x ____ ____ WIN 10 1/3 ____ LOSE ____ ____ (–12)/3 Table 4.10
Example 4.6 (part 3)c. What is the expected value, μ? Do you come out ahead?
Example 4.7Toss a fair, six-sided die twice. Let X = the number of faces that show an even number. Construct a table like Table 4.12 and calculate the mean μ and standard deviation σ of X.

Statistics4.3 Binomial Distribution (Optional)

Example 4.10A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than 10 heads? Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3,..., 15.…
Example 4.11 (part 1)a. This is a binomial problem because there is only a success or a ________, there are a fixed number of trials, and the probability of a success is.70 for each trial.
Example 4.11 (part 2)b. If we are interested in the number of students who do their homework on time, then how do we define X?
Example 4.11 (part 3)c. What values does x take on?
Example 4.11 (part 4)d. What is a failure, in words?
Example 4.11 (part 5)e. If p + q = 1, then what is q?
Example 4.11 (part 6)f. The words at least translate as what kind of inequality for the probability question P(x ____ 40)?
Example 4.13A store releases a 560-page art supply catalog. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists. What values does x take…
Example 4.14What is the probability distribution for X? Using the formulas, calculate the (i) mean and (ii) standard deviation of X. Use your calculator to find the probability that at most eight people develop the disease. Is…

Statistics4.4 Geometric Distribution (Optional)

Example 4.18 (part 1)a. Let X = the number of ________ you must ask ________ one says yes.
Example 4.18 (part 2)b. What values does X take on?
Example 4.18 (part 3)c. What are p and q?
Example 4.18 (part 4)d. The probability question is P(_______).
Example 4.20The lifetime risk of developing pancreatic cancer is about one in 78 (1.28 percent). Let X = the number of people you ask until one says he or she has pancreatic cancer. Then X is a discrete random variable with a…

Statistics4.5 Hypergeometric Distribution (Optional)

Example 4.21A candy dish contains 100 jelly beans and 80 gumdrops. Fifty candies are picked at random. What is the probability that 35 of the 50 are gumdrops? The two groups are jelly beans and gumdrops. Since the probability…
Example 4.22Suppose a shipment of 100 DVD players is known to have 10 defective players. An inspector randomly chooses 12 for inspection. He is interested in determining the probability that, among the 12 players, at most two…
Example 4.23 (part 1)a. Are you choosing with or without replacement?
Example 4.23 (part 2)b. What is the group of interest?
Example 4.23 (part 3)c. How many are in the group of interest?
Example 4.23 (part 4)d. How many are in the other group?
Example 4.23 (part 5)e. Let X = ________ on the committee. What values does X take on?
Example 4.23 (part 6)f. The probability question is P(_______).

Statistics4.6 Poisson Distribution (Optional)

Example 4.26A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day,…
Example 4.27 (part 1)a. What is the interval of interest?
Example 4.27 (part 2)b. What is the average number of times the news reporter says "uh" during one broadcast?
Example 4.27 (part 3)c. Let X = ________. What values does X take on?
Example 4.29What is the probability that an email user receives exactly 160 emails per day? What is the probability that an email user receives at most 160 emails per day? What is the standard deviation?
Example 4.30How many text messages does a text message user receive or send per hour? What is the probability that a text message user receives or sends two messages per hour? What is the probability that a text message user…
Example 4.31On May 13, 2013, starting at 4:30 p.m., the probability of low seismic activity for the next 48 hours in Alaska was reported as about 1.02 percent. Use this information for the next 200 days to find the probability…

Statistics5.2 The Uniform Distribution

Example 5.3 (part 1)a. Refer to Example 5.2. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds?
Example 5.3 (part 2)b. Find the 90th percentile for an eight-week-old baby's smiling time.
Example 5.3 (part 3)c. Find the probability that a random eight-week-old baby smiles more than 12 seconds knowing that the baby smiles more than eight seconds.
Example 5.4 (part 1)a. What is the probability that a person waits fewer than 12.5 minutes?
Example 5.4 (part 2)b. On the average, how long must a person wait? Find the mean, μ, and the standard deviation, σ.
Example 5.4 (part 3)c. Ninety percent of the time, the minutes a person must wait falls below what value? This question asks for the 90th percentile.
Example 5.5 (part 1)a. The probability that a randomly selected nine-year-old child eats a doughnut in at least two minutes is _______.
Example 5.5 (part 2)b. Find the probability that a different nine-year-old child eats a doughnut in more than two minutes given that the child has already been eating the doughnut for more than 1.5 minutes. The second question has a…With a figure
Example 5.6Find the probability that a randomly selected furnace repair requires more than two hours. Find the probability that a randomly selected furnace repair requires less than three hours. Find the 30th percentile of…

Statistics5.3 The Exponential Distribution (Optional)

Example 5.8 (part 1)a. Using the information in Example 5.7, find the probability that a clerk spends four to five minutes with a randomly selected customer.
Example 5.8 (part 2)b. Half of all customers are finished within how long? (Find the 50th percentile).
Example 5.8 (part 3)c. Which is larger, the mean or the median?
Example 5.9 (part 1)a. What is the probability that a computer part lasts more than seven years?
Example 5.9 (part 2)b. On the average, how long would five computer parts last if they are used one after another?
Example 5.9 (part 3)c. Eighty percent of computer parts last at most how long?
Example 5.9 (part 4)d. What is the probability that a computer part lasts between nine and 11 years?
Example 5.10What is m, μ, and σ? The probability that you must wait more than five minutes is _______.
Example 5.11On average, how many minutes elapse between two successive arrivals? When the store first opens, how long on average does it take for three customers to arrive? After a customer arrives, find the probability that it…
Example 5.13Find the average time between two successive calls. Find the probability that after a call is received, the next call occurs in less than 10 seconds. Find the probability that exactly five calls occur within a…

Statistics6.1 The Standard Normal Distribution

Example 6.2 (part 1)a. Suppose a person lost 10 pounds in a month. The z-score when x = 10 pounds is z = 2.5 (verify). This z-score tells you that x = 10 is ________ standard deviations to the ________ (right or left) of the mean _____…
Example 6.2 (part 2)b. Suppose a person gained three pounds, a negative weight loss. Then z = __________. This z-score tells you that x = –3 is ________ standard deviations to the __________ (right or left) of the mean.
Example 6.2 (part 3)c. Suppose the random variables X and Y have the following normal distributions: X ~ N(5, 6) and Y ~ N(2, 1). If x = 17, then z = 2. This was previously shown. If y = 4, what is z?
Example 6.3 (part 1)a. Suppose a 15-to 18-year-old male from Chile was 168 cm tall in 2009–2010. The z-score when x = 168 cm is z = _______. This z-score tells you that x = 168 is ________ standard deviations to the ________ (right or…
Example 6.3 (part 2)b. Suppose that the height of a 15-to 18-year-old male from Chile in 2009–2010 has a z-score of z = 1.27. What is the male’s height? The z-score (z = 1.27) tells you that the male’s height is ________ standard…
Example 6.4From 1984 to 1985, the mean height of 15-to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let Y = the height of 15-to 18-year-old males from 1984–1985, and y = the height of one…
Example 6.6From 1984–1985, the mean height of 15-to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let Y = the height of 15-to 18-year-old males in 1984–1985. Then Y ~ N(172.36, 6.34). About…Plus a twin to try

Statistics6.2 Using the Normal Distribution

Example 6.8 (part 1)a. Find the probability that a randomly selected student scored more than 65 on the exam.
Example 6.8 (part 2)b. Find the probability that a randomly selected student scored less than 85.
Example 6.8 (part 3)c. Find the 90th percentile, —that is, find the score k that has 90 percent of the scores below k and 10 percent of the scores above k.
Example 6.8 (part 4)d. Find the 70th percentile, —that is, find the score k such that 70 percent of scores are below k and 30 percent of the scores are above k.
Example 6.9 (part 1)a. Find the probability that a household personal computer is used for entertainment between 1.8 and 2.75 hours per day.
Example 6.9 (part 2)b. Find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment.
Example 6.10 (part 1)a. Determine the probability that a random smartphone user in the age range 13 to 55+ is between 23 and 64.7 years old.
Example 6.10 (part 2)b. Determine the probability that a randomly selected smartphone user in the age range 13 to 55+ is at most 50.8 years old.
Example 6.10 (part 3)c. Find the 80th percentile of this distribution, and interpret it in a complete sentence.
Example 6.11 (part 1)a. Calculate the interquartile range (IQR).
Example 6.11 (part 2)b. Forty percent of the ages that range from 13 to 55+ are at least what age?
Example 6.12 (part 1)a. Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.0 cm. Sketch the graph.
Example 6.12 (part 2)b. The middle 20 percent of mandarin oranges from this farm have diameters between ______ and ______.
Example 6.12 (part 3)c. Find the 90th percentile for the diameters of mandarin oranges, and interpret it in a complete sentence.Plus a twin to try

Statistics7.1 The Central Limit Theorem for Sample Means (Averages)

Example 7.1 (part 1)a. Find the probability that the sample mean is between 85 and 92.
Example 7.1 (part 2)b. Find the value that is two standard deviations above the expected value, 90, of the sample mean.
Example 7.2The length of time, in hours, it takes a group of people, 40 years old and older, to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours. A sample of size n = 50…
Example 7.3In a recent study reported Oct. 29, 2012, the mean age of tablet users is 34 years. Suppose the standard deviation is 15 years. Take a sample of size n = 100. What are the mean and standard deviation for the sample…
Example 7.4The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of 60. What are the mean and standard deviation for the sample mean number of…

Statistics7.2 The Central Limit Theorem for Sums (Optional)

Example 7.5Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7,500. Find the sum that is 1.5 standard deviations above the mean of the sums.
Example 7.6In a recent study reported Oct. 29, 2012, the mean age of tablet users is 34 years. Suppose the standard deviation is 15 years. The sample size is 50. What are the mean and standard deviation for the sum of the ages…
Example 7.7The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample size of 70. What are the mean and standard deviation for the sums? Find the…

Statistics7.3 Using the Central Limit Theorem

Example 7.8 (part 1)a. Find P( x ¯ < 2). Draw the graph.
Example 7.8 (part 2)b. Find the 90th percentile for the mean of 75 stress scores. Draw a graph.
Example 7.8 (part 3)c. Find P(Σx < 200). Draw the graph.
Example 7.8 (part 4)d. Find the 90th percentile for the total of 75 stress scores. Draw a graph.
Example 7.9 (part 1)Using the clt to find probability Find the probability that the mean excess time used by the 80 customers in the sample is longer than 20 minutes. This is asking us to find P( x ¯ > 20). Draw the graph. Suppose that…
Example 7.9Using the clt to find percentiles (part 2)Find the 95 th percentile for the sample mean excess time for a sample of 80 customers who exceed their basic contract time allowances. Draw a graph.
Example 7.10Find the median, the first quartile, and the third quartile for the sample mean time of diagnosis in the United States. Find the median, the first quartile, and the third quartile for the sum of sample times of…
Example 7.11A study was done about a medical condition that affects a certain group of people. The age range of the people was 14–61. The mean age was 30.9 years with a standard deviation of nine years. In a sample of 25 people,…