Statistics Chapter 4 Study Guide
Actual odds against:
"The actual odds against event A occurring are the ratio P(A¯)/P(A) usually expressed in the form of a: b (or " a to b"), where a and b are integers. (Reduce using the largest common factor; if a = 16 and b = 4, express the odds as 4:1 instead of 16:4.)"
Actual odds in favor:
"The actual odds in favor of event A occurring are the ratio P(A)/P(A¯), which is the reciprocal of the actual odds against that event. If the odds against an event are a: b, then the odds in favor are b: a."
Payoff odds:
"The payoff odds against event A occurring are the ratio of net profit (if you win) to the amount bet: Payoff odds against event A = (net profit) : (amount bet)"
The probability of an impossible event is:
0
The probability of an event that is certain to occur is:
1
A study of 420,095 cell phone users resulted in 135 who developed cancer of the brain or nervous system. When comparing this sample group to another group of people who did not use cell phones, it was found that there is a probability of 0.512 of getting such sample results by chance. What do you conclude?
A study of 420,095 cell phone users resulted in 135 who developed cancer of the brain or nervous system. When comparing this sample group to another group of people who did not use cell phones, it was found that there is a probability of 0.512 of getting such sample results by chance. What do you conclude?
There are 15,958,866 adults in a region. If a polling organization randomly selects 1235 adults without replacement, are the selections independent or dependent? If the selections are dependent, can they be treated as independent for the purposes of calculations?
Are the selections independent or dependent? The selections are dependent, because the selection is done without replacement. If the selections are dependent, can they be treated as independent for the purposes of calculations? Yes, because the sample size is less than 5% of the population.
In horse racing, a trifecta is a bet that the first three finishers in a race are selected, and they are selected in the correct order. Does a trifecta involve combinations or permutations? Explain.
Because the order of the first three finishers does make a difference, the trifecta involves permutations.
If A denotes some event, what does Ā denote? If P(A)=0.001, what is the value of P(Ā)?
Event Ā denotes the complement of event A, meaning that Ā consists of all outcomes in which event A does not occur. P(Ā)=0.999 (take 1 minus 0.001)
Which of the following is NOT a requirement of the Permutations Rule, nPr=n!/(n−r)!, for items that are all different?
Order is not taken into account (rearrangements of the same items are considered to be the same).
When randomly selecting an adult, A denotes the event of selecting someone with blue eyes. What do P(A) and PA represent?
P(A) represents the probability of selecting an adult with blue eyes. P(Ā) represents the probability of selecting an adult who does not have blue eyes.
Confusion of the inverse occurs when we incorrectly believe _______.
P(B|A) = P(A|B)
In a computer instant messaging survey, respondents were asked to choose the most fun way to flirt, and it found that P(D)=0.700, where D is directly in person. If someone is randomly selected, what does P(D with a line over it) represent, and what is its value?
P(D with a line over it) is the probability of randomly selecting someone who does not choose a direct in-person encounter as the most fun way to flirt. P(D with a line over it) = 0.3 (1 - 0.700)
When a man observed a sobriety checkpoint conducted by a police department, he saw 673 drivers were screened and 9 were arrested for driving while intoxicated. Based on those results, we can estimate that P(W)=0.01337, where W denotes the event of screening a driver and getting someone who is intoxicated. What does P(W with a line over it) denote, and what is its value?
P(W with a line over it) denotes the probability of screening a driver and finding that he or she is not intoxicated. P(W with a line over it) = 0.98663
A moving company has a truck filled for deliveries to eight different sites. If the order of the deliveries is randomly selected, what is the probability that it is the shortest route?
P(randomly selecting the shortest route)= 1/40320 (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
When seven basketball players are about to have a free-throw competition, they often draw names out of a hat to randomly select the order in which they shoot. What is the probability that they shoot free throws in alphabetical order? Assume each player has a different name.
P(shoot free throws in alphabetical order) = 1/5040 (take 7!)
Which of the following is NOT a requirement of the Combinations Rule, nCr=n!/r!(n−r)!, for items that are all different?
That order is taken into account (consider rearrangements of the same items to be different sequences).
In a state pick 4 lottery game, a bettor selects four numbers between 0 and 9 and any selected number can be used more than once. Winning the top prize requires that the selected numbers match those and are drawn in the same order. Do the calculations for this lottery involve the combinations rule or either of the two permutations rules? Why or why not? If not, what rule does apply?
The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement. The multiplication counting rule applies to this problem.
Assume that 2000 births are randomly selected and 1958 of the births are girls. Use subjective judgment to describe the number of girls as significantly high, significantly low, or neither significantly low nor significantly high.
The number of girls is significantly high.
A thief steals an ATM card and must randomly guess the correct five-digit pin code from a 10-key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first try?
The number of possible codes is 100000 (multiply 10 by itself 5 times) The probability that the correct code is given on the first try is 1/100000
When testing for current in a cable with seven color-coded wires, the author used a meter to test three wires at a time. How many different tests are required for every possible pairing of three wires?
The number of tests required is 35 (use combinations and permutations calculator)
Six of the 100 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability that a randomly selected item is defective?
The probability is 0.06 (take 6 divided by 100)
Express the indicated degree of likelihood as a probability value between 0 and 1. When using a computer to randomly generate the last digit of a phone number to be called for a survey, there is 1 chance in 10 that the last digit is zero.
The probability is 0.1 (1 divided by 10)
In a study of helicopter usage and patient survival, among the 56,007 patients transported by helicopter, 185 of them left the treatment center against medical advice, and the other 55,822 did not leave against medical advice. If 40 of the subjects transported by helicopter are randomly selected without replacement, what is the probability that none of them left the treatment center against medical advice?
The probability is 0.876
Among 6992 cases of heart pacemaker malfunctions, 286 were found to be caused by firmware, which is software programmed into the device. If the firmware is tested in 3 different pacemakers randomly selected from this batch of 6992 and the entire batch is accepted if there are no failures, what is the probability that the firmware in the entire batch will be accepted? Is this procedure likely to result in the entire batch being accepted?
The probability is 0.882. This procedure is likely to result in the entire batch being accepted.
You are certain to get a black or a red card when selecting cards from a shuffled deck. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
The probability is 1.
You are certain to get a red card when selecting 27 cards from a shuffled deck. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
The probability is 1.
A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt?
The probability is 1/100000 (multiply 10 by itself 5 times since we know the last 4 numbers)
In a small private school, 6 students are randomly selected from 17 available students. What is the probability that they are the six youngest students?
The probability is 1/12376 (17 x 16 x 15 x 14 x 13 x 12) (*use calculator instead)
The data represent the results for a test for a certain disease. Assume one individual from the group is randomly selected. Find the probability of getting someone who tested positive, given that he or she had the disease.
The probability is approximately 0.838 (divide the number of people that tested positive by the total number of people that actually are sick)
Refer to the sample data for pre-employment drug screening shown below. If one of the subjects is randomly selected, what is the probability that the test result is a false positive? Who would suffer from a false positive result? Why?
The probability of a false positive test result is 0.179 (divide the false positive number by the total of the numbers) Who would suffer from a false positive result? The person tested would suffer because he or she would be suspected of using drugs when in reality he or she does not use drugs.
What does P(B|A) represent?
The probability of event B occurring after it is assumed that event A has already occurred
In a genetics experiment on peas, one sample of offspring contained 412 green peas and 138 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green. Is the result reasonably close to the value of 3/4 that was expected?
The probability of getting a green pea is approximately 0.749 (divide green peas by the total numbers) Is this probability reasonably close to 3/4? Yes, it is reasonably close
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in the table. If one order is selected, find the probability of getting food that is not from Restaurant A.
The probability of getting food that is not from Restaurant A is 0.675 (add up all the numbers except for restaurant A, then divide that by the total amount of numbers.
Winning the jackpot in a particular lottery requires that you select the correct three numbers between 1 and 48 and, in a separate drawing, you must also select the correct single number between 1 and 30. Find the probability of winning the jackpot
The probability of winning the jackpot is 1/518880 (48 x 48 x 48, then multiply that by 30)
Winning the jackpot in a particular lottery requires that you select the correct five numbers between 1 and 42 and, in a separate drawing, you must also select the correct single number between 1 and 27. Find the probability of winning the jackpot.
The probability of winning the jackpot is = 1/22968036 (42 x 42 x 42 x 42 x 42, then multiply that # by 27)
A study of the effect of seatbelt use in head-on passenger car collisions found that drivers using a seatbelt had a 64.1% survival rate, while drivers not using a seatbelt had a 41.5% survival rate. If seatbelts have no effect on survival rate, there is less than a 0.0001 chance of getting these results. What do you conclude?
The probability shows that the sample results could not have easily occurred by chance. It appears that there is sufficient evidence to conclude that seatbelts do have an effect on survival rate.
A study addressed the issue of whether pregnant women can correctly predict the gender of their baby. Among 104 pregnant women, 57 correctly predicted the gender of their baby. If pregnant women have no such ability, there is a 0.327 probability of getting such sample results by chance. What do you conclude?
The probability shows that the sample results could have easily occurred by chance. It appears that there is not sufficient evidence to conclude that pregnant women can correctly predict the gender of their baby.
In a test of a sex-selection technique, results consisted of 299 female babies and 7 male babies. Based on this result, what is the probability of a female being born to a couple using this technique? Does it appear that the technique is effective in increasing the likelihood that a baby will be a female?
The probability that a female will be born using this technique is approximately 0.977 (add up both totals of babies then divide the # of females from that)
Refer to the sample data for polygraph tests shown below. If one of the test subjects is randomly selected, what is the probability that the subject is not lying? Is the result close to the probability of 0.495 for a negative test result? Consider the result close if the absolute difference is less than 0.050.
The probability that a randomly selected polygraph test subject was not lying is 0.485 (divide the total of the "did not lie" column by the total numbers) Is the result close to the probability, rounded to three decimal places, of 0.495 for a negative test result? Consider the result close if the absolute difference is less than 0.050. Yes, because there is less than a 0.050 absolute difference between the probability of a true response and the probability of a negative test result.
A research center poll showed that 78% of people believe that it is morally wrong to not report all income on tax returns. What is the probability that someone does not have this belief?
The probability that someone does not believe that it is morally wrong to not report all income on tax returns is 0.22 (100 minus 78, then divide by 100)
Let event A=subject is telling the truth and event B=polygraph test indicates that the subject is lying. Use your own words to translate the notation P(B|A) into a verbal statement.
The probability that the polygraph indicates lying given that the subject is actually telling the truth.
Find the indicated complement. A certain group of women has a 0.01% rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have red/green color blindness?
The probability that the woman selected does not have red/green color blindness is 0.9999 (1 - 0.01)
With a short time remaining in the day, a delivery driver has time to make deliveries at 7 locations among the 9 locations remaining. How many different routes are possible?
There are 181440181440 possible different routes. (9 x 8 x 7 x 6 x 5 x 4 x 3)
If radio station call letters must begin with either K or W and must include either two or three additional letters, how many different possibilities are there?
There are 36504 different possibilities (2x26x26x26 add it to 2x26x26)
A weather forecasting website indicated that there was a 60% chance of rain in a certain region. Based on that report, which of the following is the most reasonable interpretation?
There is a 0.60 probability that it will rain somewhere in the region at some point during the day.
When randomly selecting adults, let M denote the event of randomly selecting a male and let B denote the event of randomly selecting someone with blue eyes. What does P(M|B) represent? Is P(M|B) the same as P(B|M)?
What does P(M|B) represent? The probability of getting a male, given that someone with blue eyes has been selected. Is P(M|B) the same as P(B|M)? No, because P(B|M) represents the probability of getting someone with blue eyes, given that a male has been selected.
To the right are the outcomes that are possible when a couple has three children. Assume that boys and girls are equally likely, so that the eight simple events are equally likely. Find the probability that when a couple has three children, there are exactly 3 boys.
What is the probability of exactly 3 boys out of three children? 0.125 (# of boys from the question divided by total from all the columns)
The probability of an event is:
a fraction or decimal number between 0 and 1 inclusive.
Men have XY (or YX) chromosomes and women have XX chromosomes. X-linked recessive genetic diseases (such as juvenile retinoschisis) occur when there is a defective X chromosome that occurs without a paired X chromosome that is not defective. Represent a defective X chromosome with lowercase x, so a child with the xY or Yx pair of chromosomes will have the disease and a child with XX or XY or YX or xX or Xx will not have the disease. Each parent contributes one of the chromosomes to the child. Complete parts a through d below.
a) If a father has the defective x chromosome and the mother has good XX chromosomes, what is the probability that a son will inherit the disease? 0 b) If a father has the defective x chromosome and the mother has good XX chromosomes, what is the probability that a daughter will inherit the disease? 0 c) If a mother has one defective x chromosome and one good X chromosome and the father has good XY chromosomes, what is the probability that a son will inherit the disease? 0.5 d. If a mother has one defective x chromosome and one good X chromosome and the father has good XY chromosomes, what is the probability that a daughter will inherit the disease? 0
Each of two parents has the genotype blue/green, which consists of the pair of alleles that determine eye color, and each parent contributes one of those alleles to a child. Assume that if the child has at least one blue allele, that color will dominate and the child's eye color will be blue.
a) List the possible outcomes: blue/blue, blue/green, green/blue, green/green b) The probability that a child of these parents will have the green/green genotype is 0.25. c) The probability that the child will have blue eye color is 0.75.
A study on the enhancing effect of coffee on long-term memory found that 35 participants given 200 mg of caffeine performed better on a memory test 24 hours later compared to the placebo group that received no caffeine.
a) The probability shows that the sample results could not have easily occurred by chance. It appears that there is sufficient evidence to conclude that 200 mg of caffeine does have an effect on memory. b. The probability shows that the sample results could have easily occurred by chance. It appears that there is not sufficient evidence to conclude that the effects from the 300 mg treatment and the 200 mg treatment are different.
On their first date, Kelly asks Mike to guess the date of her birth, not including the year. Complete parts a through c below.
a) What is the probability that Mike will guess correctly? 1/365 b) Would is be unlikely for him to guess correctly on his first try? Yes, it is unlikely for Mike to guess correctly on his first try, as the probability of a correct guess is very low. c) If you were Kelly, and Mike did guess correctly on his first try, would you believe his claim that he made a lucky guess, or would you be convinced that he already knew when you were born? Mike already knew, as the probability of a correct guess is very low.
In an experiment, college students were given either four quarters or a $1 bill and they could either keep the money or spend it on gum. The results are summarized in the table. Complete parts (a) through (c) below.
a. Find the probability of randomly selecting a student who spent the money, given that the student was given four quarters. 0.723 (add up the quarter category and then divide the number of ppl who spend the money by that number) b. Find the probability of randomly selecting a student who kept the money, given that the student was given four quarters. 0.277 c. What do the preceding results suggest? A student given four quarters is more likely to have spent the money.
Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 24% of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below.
a. Find the probability that both generators fail during a power outage. 0.0576 (0.24 x 0.24) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed. 0.9424 (1 - 0.0576) Is that probability high enough for the hospital? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. No, because both generators fail about 6% of the time they are needed. Given the importance of the hospital's needs, the reliability should be improved.
A corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with five different members. There are 12 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below.
a. How many different ways can the officers be appointed? There are 11880 different ways to appoint the officers. ( use the calculator with is order important marked yes) b. How many different ways can the committee be appointed? There are 792 different ways to appoint the committee. (use calculator with is order important marked no) c. What is the probability of randomly selecting the committee members and getting the five youngest of the qualified candidates? P(getting the five youngest of the qualified candidates)= 1/792
A corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with four different members. There are 13 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below.
a. How many different ways can the officers be appointed? There are 17160 different ways to appoint the officers. (use the calculator with the is order important marked yes) b. How many different ways can the committee be appointed? There are 715 different ways to appoint the committee. (use the calculator with the is order important marked no) c. What is the probability of randomly selecting the committee members and getting the four youngest of the qualified candidates? P(getting the four youngest of the qualified candidates)= 1/715
A corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with three different members. There are 11 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below.
a. How many different ways can the officers be appointed? There are 79207920 different ways to appoint the officers. (use the calculator with is order important marked yes) b. How many different ways can the committee be appointed? There are 165 different ways to appoint the committee. (use the calculator with the is order important marked no) c. What is the probability of randomly selecting the committee members and getting the three youngest of the qualified candidates? P(getting the three youngest of the qualified candidates)= 1/165
Use the following results from a test for marijuana use, which is provided by a certain drug testing company. Among 141 subjects with positive test results, there are 29 false positive results. Among 150 negative results, there are 2 false negative results. Complete parts (a) through (c). (Hint: Construct a table.)
a. How many subjects were included in the study? 291 b. How many subjects did not use marijuana? 177 c. What is the probability that a randomly selected subject did not use marijuana? 0.608 (177 divided by 291)
A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of a specific drug involved only 7 subjects. Assume that we want to treat 7 healthy humans with this new drug and we have 9 suitable volunteers available. Complete parts (a) through (c) below.
a. If the subjects are selected and treated in sequence, so that the trial is discontinued if anyone displays adverse effects, how many different sequential arrangements are possible if 7 people are selected from the 9 that are available? Choose the correct answer below. 181,440 (9 x 8 x 7 x 6 x 5 x 4 x 3 ) b. If 7 subjects are selected from the 9 that are available, and the 7 selected subjects are all treated at the same time, how many different treatment groups are possible? There are 36 different treatment groups possible. (use the calculator with is order important marked no) c. If 7 subjects are randomly selected and treated at the same time, what is the probability of selecting the 7 youngest subjects? P(selecting the 7 youngest subjects) = 1/36
A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of a specific drug involved only 8 subjects. Assume that we want to treat 8 healthy humans with this new drug and we have 13 suitable volunteers available. Complete parts (a) through (c) below.
a. If the subjects are selected and treated in sequence, so that the trial is discontinued if anyone displays adverse effects, how many different sequential arrangements are possible if 8 people are selected from the 13 that are available? Choose the correct answer below. C. 51,891,840 (13 x 12 x 11 x 10 x 9 x 8 x 7 x 6) b. If 8 subjects are selected from the 13 that are available, and the 8 selected subjects are all treated at the same time, how many different treatment groups are possible? There are 1287 different treatment groups possible. (use the calculator with the is order important marked no) c. If 8 subjects are randomly selected and treated at the same time, what is the probability of selecting the 8 youngest subjects? P(selecting the 8 youngest subjects)=1/1287
You want to obtain cash by using an ATM, but it's dark and you can't see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first. Complete parts (a) through (c).
a. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The probability is 1/4. (2 x 2) b. What is the probability of randomly selecting the card's position and finding that it is incorrectly inserted on the first attempt, but it is correctly inserted on the second attempt? (Assume that the same position used for the first attempt could also be used for the second attempt.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The probability is 3/16. (3/4 x 1/4) c. How many random selections are required to be absolute sure that the card works because it is inserted correctly? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. B. This is a trick question. There is no finite number of attempts, because it is possible to get the wrong position every time.
The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that a student's alarm clock has a 16.7% daily failure rate. Complete parts (a) through (d) below.
a. What is the probability that the student's alarm clock will not work on the morning of an important final exam? 0.167 (16.7 divided by 100) b. If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important final exam? 0.02789 (0.167 x 0.167) c. What is the probability of not being awakened if the student uses three independent alarm clocks? 0.00466 (0.167 x 0.167 x 0.167) d. Do the second and third alarm clocks result in greatly improved reliability? Yes, because total malfunction would not be impossible, but it would be unlikely.
When using the _______ always be careful to avoid double-counting outcomes.
addition rule
The conditional probability of B given A can be found by _______.
assuming that event A has occurred, and then calculating the probability that event B will occur
The ______________ of event A, denoted by Ā, consists of all outcomes in which event A does not occur.
complement
A _______ is any event combining two or more simple events.
compound event
A _______ probability of an event is a probability obtained with knowledge that some other event has already occurred.
conditional
Events that are _______ cannot occur at the same time.
disjoint
The classical approach to probability requires that the outcomes are _______.
equally likely.
An ___________ is any collection of results or outcomes of a procedure.
event
Selections made with replacement are considered to be _______.
independent
Two events A and B are _______ if the occurrence of one does not affect the probability of the occurrence of the other.
independent
If the order of the items selected matters, then we have a _______.
permutation problem
P(A)+P(Ā)= 1 is one way to express the_______.
rule of complementary events
The ____________________ for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.
sample space
The _______ for a procedure consists of all possible simple events or all outcomes that cannot be broken down any further.
sample space
A __________________ is an outcome or an event that cannot be further broken down into simpler components.
simple event
As a procedure is repeated again and again, the relative frequency of an event tends to approach the actual probability. This is known as _______.
the law of large numbers
For a sequence of events in which the first event can occur n1 ways, the second event can occur n2 ways, the third event can occur n3 ways, and so on, the events together can occur a total of n1•n2•n3•... ways. This is called _______.
the multiplication counting rule.
Notation: P(A) =
the probability of event A
Notation: P(Ā) =
the probability that event A does not occur
P(A or B) indicates _______.
the probability that in a single trial, event A occurs, event B occurs, or they both occur.
A picture of line segments branching out from one starting point illustrating the possible outcomes of a procedure is called a _______.
tree diagram
A combination lock uses three numbers between 1 and 78 with repetition, and they must be selected in the correct sequence. Is the name of "combination lock" appropriate? Why or why not?
No, because the multiplication counting rule would be used to determine the total number of combinations.
If you know the names of the remaining eight students in the spelling bee, what is the probability of randomly selecting an order and getting the order that is used in the spelling bee?
P(selecting the correct spelling bee order) = 1/40320 (multiply 8x7x6x5x4x3x2x1)
A presidential candidate plans to begin her campaign by visiting the capitals in 4 of 43 states. What is the probability that she selects the route of four specific capitals?
P(she selects the route of four specific capitals)= 1/2961840 (43 x 42 x 41 x 40)
Which of the following values cannot be probabilities? 1, √2, 0, 3/5, 5/3, 1.52, 0.09, −0.42
√2, -0.42, 5/3, 1.52
