Test 2

Suppose that E and F are two events and that P(E and F) = 0.1 and P(E) = 0.2. What is P(F|E)​?

0.5 because 0.1 / 0.2 = 0.5 If E and F are any two​ events, then the probability of event F​ occurring, given the occurrence of event​ E, is found by dividing the probability of E and F by the probability of E. P(F|E) = P(E and F) / P(E)

Find the value of the permutation of 19P0

19P0 = 1 use formula: nPr = n! / (n - r)! 19! / (19 - 0)! --> 19! / 19! = (use calculator)

Find the value of the permutation of 4P2

4P2 = 12 use formula: nPr = n! / (n - r)! 4! / (4 - 2)! --> 4! / 2! = 12 (use calculator)

What does it mean for an event to be​ unusual? Why should the cutoff for identifying unusual events not always be​ 0.05?

An event is unusual if it has a low probability of occurring. The choice of a cutoff should consider the context of the problem.

The probability that a randomly selected individual in a country earns more than​ $75,000 per year is 7.5​%. The probability that a randomly selected individual in the country earns more than​ $75,000 per​ year, given that the individual has earned a​ bachelor's degree, is 21.5​%. Are the events​ "earn more than​ $75,000 per​ year" and​ "earned a​ bachelor's degree"​ independent?

No

Data Values: .25, .25, .15, .15, - .3, .2 Determine why it is not a probability model?

This is not a probability model because at least one probability is less than 0. note: one probability is - .3

True or False​: In a probability​ model, the sum of the probabilities of all outcomes must equal 1.

True

x-values: y-values 0: 0.31 1: 0.28 2: 0.16 3: 0.11 4: 0.14 Is the distribution a discrete probability​ distribution?

Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and​ 1, inclusive.

Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Three cards are selected from a standard​ 52-card deck without replacement. The number of clubs selected is recorded.

​No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.

Determine if the following probability experiment represents a binomial experiment. A random sample of 80 professional athletes is​ obtained, and the individuals selected are asked to state their ages.

​No, this probability experiment does not represent a binomial experiment because the variable is​ continuous, and there are not two mutually exclusive outcomes.

Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 160 randomly selected​ individuals, with the number of individuals responding favorably recorded.

​Yes, because the experiment satisfies all the criteria for a binomial experiment.

About 12​% of the population of a large country is allergic to pollen. If two people are randomly​ selected, what is the probability both are allergic to pollen​? What is the probability at least one is allergic to pollen​? Assume the events are independent. ​(a) The probability that both will be allergic to pollen is ??? (b) The probability that at least one person is allergic to pollen is ???

(a) .0144 because .12 x .12 = .0144 note: multiply the probability of one person, twice (b) .2256 ​P(at least 1 allergic​) = 1 −​ P(neither allergic) 1 - (.88 x .88) --> 1 - .7744 = .2256

(a) probability that the card drawn from a standard​ 52-card deck is a queen is ??? (b) The probability that the card drawn from a standard​ 52-card deck is a queen​, given that this card is court​, is ???

(a) .077 because 4 / 52 = .077 note: there are 4 queens in a deck (b) .333 because 4 / 12 = .333 note: there are 12 quarts in a deck (4 of each: kings, queens, and jacks)

A test to determine whether a certain antibody is present is 99.7​% effective. This means that the test will accurately come back negative if the antibody is not present​ (in the test​ subject) 99.7​% of the time. The probability of a test coming back positive when the antibody is not present​ (a false​ positive) is 0.003. 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) .9881 because .997^4 = .9881 (b) .0119 because 1 - .9881 = .0119 P(at least one positive) = 1 - (all tests are negative)

(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) 0 (zero) (b) no note: The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the​ experiment, as shown below. Just because the event is not​ observed, does not mean that the event is impossible.

Determine whether the events E and F are independent or dependent. Justify your answer. ​(a) E: A person living at least 70 years. F: The same person regularly handling venomous snakes. (b) E: A randomly selected person coloring her hair black. ​F: A different randomly selected person coloring her hair blond. (c) E: The rapid spread of a cocoa plant disease. ​F: The price of chocolate.

(a) E and F are dependent because regularly handling venomous snakes can affect the probability of a person living at least 70 years. (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.

A probability experiment is conducted in which the sample space of the experiment is S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}​, event E={2, 3, 4, 5} and event G={6, 7, 8, 9}. Assume that each outcome is equally likely. (a) List the outcomes in E and G. (b) Are E and G mutually​ exclusive?

(a) E and G = ​{} note: they share no variables in common (b) Yes, because the events E and G have no outcomes in common.

Match the linear correlation coefficient to the scatter diagram. The scales on the​ x- and​ y-axis are the same for each scatter diagram. (a) r = −0.969​ (b) r = −1​ (c) r = - 0.049

(a) Graph 2: dots go negative, not exact line (b) Graph 1: dots go in a negative line exactly (c) Graph 3: barley noticable trend/line

For the accompanying data​ set, (a) draw a scatter diagram of the​ data, (b) by​ hand, compute the correlation​ coefficient, and​ (c) determine whether there is a linear relation between x and y. (a) Draw a scatter diagram of the data. (b) By​ hand, compute the correlation coefficient. The correlation coefficient is ??? (c) Because the correlation coefficient is ??? and the absolute value of the correlation​ coefficient, ??? is ??? than the critical value for this data​ set, ???, ??? linear relation exists between x and y.

(a) Look at points and plot them, choose best graph (b) - .496 (use calculator) (c) negative, .496, not greater, .878, no - note: it asks for the absolute value, never a negative - note: if correlation coefficient < critical value then there is no linear relation

For the accompanying data​ set, (a) draw a scatter diagram of the​ data, (b) by​ hand, compute the correlation​ coefficient, and​ (c) determine whether there is a linear relation between x and y. (a) Draw a scatter diagram of the data. (b) By​ hand, compute the correlation coefficient. The correlation coefficient is ??? (c) Because the correlation coefficient is ??? and the absolute value of the correlation​ coefficient, ??? is ??? than the critical value for this data​ set, ???, ??? linear relation exists between x and y.

(a) Look at points and plot them, choose best graph (b) .943 (use calculator) (c) positive, .943, greater, .878, a positive - find critical value by looking at given table and looking at the y-value for the number of variables

Data Values: .2, .1, .25, .25, .15 (a) Is the table above an example of a probability​ model? (b) What do we call the outcome "blue​"?

(a) No​, because the probabilities do not sum to 1. note: add up probabilities; this equaled .95 not 1 (b) Impossible note: blue's probability is 0

​(a) If the pediatrician wants to use height to predict head​ circumference, determine which variable is the explanatory variable and which is the response variable. ​(b) Draw a scatter diagram. Which represents the​ data? ​(c) Compute the linear correlation coefficient between the height and head circumference of a child. ​(d) Does a linear relation exist between height and head​ circumference? (e) The new linear correlation coefficient is r = ??? The conversion to centimeters had ???

(a) The explanatory variable is height and the response variable is head circumference. (b) Graph Description: Basic positive trend, match points to table (c) r = .893 (use calculator) (d) ​Yes, the variables height and head circumference are positively associated because r is positive and the absolute value of the correlation coefficient is greater than the critical​ value, .707 (e) .893, has no effect on r note: conversion of values will not affect correlation coefficient

Researchers initiated a​ long-term study of the population of American black bears. One aspect of the study was to develop a model that could be used to predict a​ bear's weight​ (since it is not practical to weigh bears in the​ field). One variable thought to be related to weight is the length of the bear. The accompanying data represent the lengths and weights of 12 American black bears. (a) Which variable is the explanatory variable based on the goals of the​ research? (b) ​Draw a scatter diagram of the data. (c) The linear correlation coefficient between weight & length is ??? (d) The variables weight of the bear and length of the bear are ???associated because r is ??? and the absolute value of the correlation​ coefficient, ??? is ??? than the critical​ value, ???

(a) The length of the bear note: the length will determine the weight (response) (b) Look at basic trend, and discover the highest and lowest points to compare to choices (c) .718 (use calculator) (d) positively, positive, .718, greater, .576

​(a) Is the number of free-throw attempts before the first shot is made discrete or​ continuous? (b) Is the distance a baseball travels in the air after being hit discrete or​ continuous?

(a) The random variable is discrete. The possible values are x=​0, ​1, ​2,... (b) The random variable is continuous. The possible values are d>0.

​(a) Is the number of people with blood type A in a random sample of 31 people discrete or​ continuous? (b) Is the time it takes for a light bulb to burn out discrete or​ continuous?

(a) The random variable is discrete. The possible values are x=​0, ​1, ​2,...31 (b) The random variable is continuous. The possible values are t>0.

(a) List all the combinations of five objects x, y, z, s, and t taken two at a time. (b) What is 5C2​?

(a) xy, xz, xs, xt, yz, ys, yt, zs, zt, st note: A combination is a​ collection, without regard to​ order, of n distinct objects without repetition. (b) 10 use formula: nCr = n! / r!(n - r)! 5! / 2! (5 - 3)! --> 5! / 2!(2!) = 10

(a) List all the permutations of four objects x, y, z, and s taken two at a time without repetition. (b) What is 4P2​?

(a) xy, xz, xs, yx, yz, ys, zx, zy, zs, sx, sy, sz (b) 12

An engineer wants to determine how the weight of a​ car, x, affects gas​ mileage, y. The following data represent the weights of various cars and their miles per gallon. Weight (x) : Miles Per Gallon (y) 2510: 23.2 2940: 19.2 3295: 19.9 3875: 16.6 4225: 10.5 ​(a) Find the​ least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. (b) Interpret the slope and​ intercept, if appropriate. (c) Predict the miles per gallon of car C (3295) and compute the residual. Is the miles per gallon of this car above average or below average for cars of this​ weight? The predicted value is ??? miles per gallon The residual is ??? miles per gallon. Is the value above or below​ average? (d) Which of the following represents the data with the residual shown?

(a) y = - .00639x + 39.4 (use calculator) (b) The slope indicates the mean change in miles per gallon for an increase of 1 pound in weight. It is not appropriate to interpret the​ y-intercept because it does not make sense to talk about a car that weighs 0 pounds. (c) 18.34, 1.56, it is above average note: plug 3295 into the discovered least-squares regression line note: 19.9 - 18.34 = 1.56 (observed - predicted) note: "it" refers to the value in the table (d) Input table points into L1 and L2, input least-squares regression line into y= and graph both to compare to choices

(a) Find the​ least-squares regression line treating the number of​ absences, x, as the explanatory variable and the final​ grade, y, as the response variable. ​(b) Interpret the slope and​ y-intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the observed final grade above or below average for this number of​ absences? The predicted final grade is ??? This observation has a residual of ??? which indicates that the final grade is ??? average. (d) Draw the​ least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the​ least-squares regression line to predict the final grade for a student who has missed 15 class​ periods?

(a) y = - 3.085x +89.335 (b) Slope: For every day​ absent, the final grade falls by 3.085 on average. Y-Int: For zero days​ absent, the final score is predicted to be 89.335. (c) 73.9, - .1, below note: plug 5 in found equation and then do observed value minus predicted value (what we just found) (d) Graph both using calculator (e) No, 15 missed classes is outside of the scope model

A golf ball is selected at random from a golf bag. If the golf bag contains 6 type A​ balls, 8 type B​ balls, and 7 type C​ balls, find the probability that the golf ball is not a type A ball. The probability that the golf ball is not a type A ball is ???

.714 note: add up everything but the A values and divide that by the total occurrences (8 + 7) / (6 + 7 +8) = .714

A probability experiment is conducted in which the sample space of the experiment is S={5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}​, event F = {6, 7, 8, 9, 10, 11}​, and event G = {10, 11, 12, 13}. Assume that each outcome is equally likely. (a) List the outcomes in F or G. (b) Find P(F or G) by counting the number of outcomes in F or G. (c) Determine P(F or G) using the general addition rule.

(a) {6, 7, 8, 9, 10, 11, 12, 13} note: "or" implies that you list the variables of all sets without repeating (b) .667 note: find the number of values in the set F or G and divide that by the total set (S) which is 8/12 = .667 (c) P(F or G) = .5 + .333 − .167 = .666 ​use formula: P(F or ​G)=​ P(F) + ​P(G) − ​P(F and​ G) P(F) = variables in F / total variable count (S) P(F and G) = variables in common / total variable

A probability experiment is conducted in which the sample space of the experiment is S= {3,4,5,6,7,8,9,10,11,12,13,14}. Event E= {4,5,6,7,8,9} and event F= {8,9,10,11}. List the outcomes in E and F. Are E and F mutually​ exclusive? (a) List the outcomes in E and F (b) Are E and F mutually​ exclusive?

(a) {8,9} note: list values which they have in common (b) No. E and F have outcomes in common note: Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events.

Find probability​ P(E or​ F) if E and F are mutually​ exclusive, ​P(E) = 0.25​, and ​P(F) = 0.47. The probability​ P(E or​ F) is ???

.72 note: Use the general addition rule to find the probability. The rule states that for any two events E and​ F, P(E or F) = ​P(E) +​ P(F) − ​P(E and​ F) .25 + .47 - 0 = .72 - we know P(E and F) are 0 because they are mutually exclusive and therefore have nothing in common

Suppose that E and F are two events and that N(E and F)=410 and N(E)=510. What is P(F|E)​?

.804 because 501 / 401 = .804

For the month of June in a certain​ city, 95​% of the days are cloudy. Also in the month of June in the same​ city, 23​% of the days are cloudy and foggy. What is the probability that a randomly selected day in June will be foggy if it is cloudy​?

.242 because .23 / .95 = .242 note: P(F|E) = P(E and F) / P(E) so the probability that it is cloudy and foggy divided by probability that it is cloudy

Find probability of the indicated event if ​P(E) = 0.30 and ​P(F) = 0.35. Find​ P(E and​ F) if​ P(E or ​F) = 0.40 ​P(E and ​F) = ???

.25 note: Use the general addition rule to find the probability. The rule states that for any two events E and​ F, P(E or F) = ​P(E) +​ P(F) − ​P(E and​ F) .40 = .30 + .35 - X --> .40 = .65 - X --> X = .25

Suppose events E and F are​ independent, ​P(E) = 0.6​, and ​P(F) = 0.7. What is the P(E and F)​? The probability P(E and F) is ???

.42 note: multiply the values due to them being independent so .6 x .7 = .42

A golf ball is selected at random from a golf bag. If the golf bag contains 7 orange ​balls, 4 green ​balls, and 13 yellow ​balls, find the probability of the following event. The golf ball is orange or green. The probability that the golf ball is orange or green is ???

.458 note: add up the values of orange and green and divide it by the total occurrences 7 + 4 / 24 = .458

Find the probability ​P(E^c​) if ​P(E)=0.38. The probability ​P(E^c​) is ???

.62 note: if E represents any event and E^c represents the complement of​ E, then the probability PE^c is given by the formula below. ​P(E^c​) = 1 −​ P(E) 1 - .38 = .62

Find probability of the indicated event if ​P(E) = 0.35 and ​P(F) = 0.35. Find​ P(E or​ F) if​ P(E and ​F) = 0.05. ​P(E or ​F) = ???

.65 note: Use the general addition rule to find the probability. The rule states that for any two events E and​ F, P(E or F) = ​P(E) +​ P(F) − ​P(E and​ F) .35 + .35 - .05 = .65

Find the value of the permutation of 4P4

4P4 = 24 use formula: nPr = n! / (n - r)! 4! / (4 - 4)! --> 4! / 0! = 24 (use calculator)

Notation P(F|E) means the probability of event ??? given event ???

F, E

Determine if the following statement is true or false. When two events are​ disjoint, they are also independent.

False note: Two events are disjoint if they have no outcomes in common. Independence means that one event occurring does not affect the probability of the other event occurring.​ Therefore, knowing two events are disjoint means that the events are not independent.

Match the linear correlation coefficient to the scatter diagram. r= - 0.93

Graph Description: Dots have a trend of going down as you move right (negative) and follows a line shape very well

x-values: y-values 10: 0.36 20: 0.34 30: 0.31 40: 0.15 50: -0.16 Is the distribution a discrete probability​ distribution?

No​, because each probability is not between 0 and 1, inclusive.

x-values: y-values 0: .5 1: .5 2: .5 3: .5 4: .5 Is the distribution a discrete probability​ distribution?

No​, because the sum of the probabilities is not equal to 1.

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 15​, p = 0.7​, x = 12 What is P(12)?

P(12) = .17 use formula: P(x) = nCx p^x (1 − p)^n−x P(12) = 15C12 (.7)^12 (1 - .7)^15 - 12 --> P(12) = [ 15! / 12! (15 - 12)! ] (.7)^12 (.3)^3 = .17

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 20​, p = 0.01​, x = 2 What is P(2)?

P(2) = .0159 use formula: P(x) = nCx p^x (1 − p)^n−x P(2) = 20C2 (0.01)^2 (1 - 0.01)^20 - 2 P(2) = [ 20! / 2! (20 - 2)! ] (0.01)^2 (.99)^18 = .0159

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 7​, p = 0.45​, x = 6 What is P(6)?

P(6) = .032 use formula: P(x) = nCx p^x (1 − p)^n−x P(6) = [ 7! / 6! (7 - 6)! ] (.45)^6 (.55)^1 = .032

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= {5, 8}.

P(E) = .2 Out of the values, 5 and 8 occur once each. Therefore, there is a 2/10 chance you will get one of those values

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=​"an even number less than 10​."

P(E) = .4 Values less than 10 and even are: 2, 4, 6, 8 which is 4/10 given in the set and therefore .4

Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is​ linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following. Graph Description: Dots are scattered greatly and possibly show a positive correlation but it does not follow a line. a. Do the two variables have a linear​ relationship? b. If the relationship is linear do the variables have a positive or negative​ association?

a. The data points do not have a linear relationship because they do not lie mainly in a straight line. b. The relationship is not linear

Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is​ linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following. Graph Description: Dots generally increase as you move right, a line could be drawn in the path a. Do the two variables have a linear​ relationship? b. If the relationship is linear do the variables have a positive or negative​ association?

a. The data points have a linear relationship because they lie mainly in a straight line b. The two variables have a positive association.

The word or in probability implies that we use the ??? Rule.

addition

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