Sophia Milestone 3 exam

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Capella University *

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2001

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Statistics

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Jan 9, 2024

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26/27 that's 96% RETAKE 26 questions were answered correctly . 1 question was answered incorrectly . 1 Using the Venn Diagram below, what is the conditional probability of event A occurring, assuming that event B has already occurred [P(A| B)]?  0.05  0.22  0.10  0.71 RATIONALE To get the probability of A given B has occurred, we can use the following conditional formula: The probability of A and B is the intersection, or overlap, of the Venn diagram, which is 0.1. The probability of B is all of Circle B, or 0.1 + 0.35 = 0.45. CONCEPT Conditional Probability Report an issue with this question 2 Mark noticed that the probability that a certain player hits a home run in a single game is 0.175. Mark is interested in the variability of the number of home runs if this player plays 200 games.

If Mark uses the normal approximation of the binomial distribution to model the number of home runs, what is the standard deviation for a total of 200 games? Answer choices are rounded to the hundredths place.  28.88  5.37  5.92  0.14 RATIONALE In this situation, we know: n = sample size = 200 p = success probability = 0.175 We can also say that q, or the complement of p, equals: q = 1 - p = 1 - 0.175 = 0.825 The standard deviation is equivalent to the square root of the variance. First, find the variance. The variance is equivalent to n*p*q: Now, take the square root to find the standard deviation: CONCEPT Normal Distribution Approximation of the Binomial Distribution Report an issue with this question 3 What is the probability of NOT rolling a four when rolling a six sided die? 

   RATIONALE Recall that the probability of a complement, or the probability of something NOT happening, can be calculated by finding the probability of that event happening, and then subtracting from 1. Note that the probability of rolling a four would be 1/6. So the probability of NOT rolling a four is equivalent to: CONCEPT Complement of an Event Report an issue with this question 4 Three hundred students in a school were asked to select their favorite fruit from a choice of apples, oranges, and mangoes. This table lists the results. Boys Gi Apple 66 4 Orange 52 4 Mango 40 5 If you were to choose a boy from the group, what is the probability that mangoes are his favorite fruit? Answer choices are rounded to the hundredths place.  0.39  0.13

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 0.75  0.25 RATIONALE The probability of a person picking mangoes as his favorite fruit given he is a boy is a conditional probability. We can use the following formula: Remember, to find the total number of boys, we need to add all values in this column: 66 + 52 + 40 = 158. CONCEPT Conditional Probability and Contingency Tables Report an issue with this question 5 Kate was trying to decide which type of frozen pizza to restock based on popularity: pepperoni pizza or sausage pizza. After studying the data, she noticed that pepperoni flavors sold best on the weekdays and on the weekends, but not best overall. Which paradox has Kate encountered?  Benford's Law  Simpson's Paradox  False Negative  False Positive RATIONALE This is an example of Simpson's paradox, which is when the trend overall is not the same that is examined in smaller groups. Since the sale of pepperoni flavors on weekend/weekdays is larger but this trend changes when looking at overall sales, this is a reversal of the trend.

CONCEPT Paradoxes Report an issue with this question 6 Ryan is playing a multiplication game with a pile of 26 cards, each with a number on them. Each turn, he flips over two of the cards, and has to multiply the numbers. How many possible outcomes are there on Ryan's first turn flipping two cards?  650  676  26  52 RATIONALE We can use the general counting principle and note that for each step, we simply multiply all the possibilities at each step to get the total number of outcomes. If we assume that the numbers are 1 - 26, then the overall number of outcomes is: Note that once a number is chosen it cannot be chosen again. So the number of possible outcomes for the first card would be 26 since they could choose any card number 1 through 26. However, the second card chosen would only have 25 possible outcomes since the first card has already been drawn. CONCEPT Fundamental Counting Principle Report an issue with this question 7

Annika was having fun playing poker. She needed the next two cards dealt to be diamonds so she could make a flush (five cards of the same suit). There are 15 cards left in the deck, and five are diamonds. What is the probability that the two cards dealt to Annika (without replacement) will both be diamonds? Answer choices are in percentage format, rounded to the nearest whole number.  10%  13%  33%  29% RATIONALE If there are 15 cards left in the deck with 5 diamonds, the probability of being dealt 2 diamonds if they are dealt without replacement means that we have dependent events because the outcome of the first card will affect the probability of the second card. We can use the following formula: The probability that the first card is a diamond would be 5 out of 15, or . The probability that the second card is a diamond, given that the first card was also a diamond, would be because we now have only 14 cards remaining and only 4 of those cards are diamond (since the first card was a diamond). So we can use these probabilities to find the probability that the two cards will both be diamonds: CONCEPT "And" Probability for Dependent Events Report an issue with this question

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8 A credit card company surveys 125 of its customers to ask about satisfaction with customer service. The results of the survey, divided by gender, are shown below. Males Extremely Satisfied 25 Satisfied 21 Neutral 13 Dissatisfied 9 Extremely Dissatisfied 2 If a survey is selected at random, what is the probability that the person is a female with neutral feelings about customer service? Answer choices are rounded to the hundredths place.  0.29  0.5  0.13  0.81  0.19 RATIONALE If we want the probability of selecting a survey that is from a female who marked "neutral feelings," we just need to look at the box that is associated with both categories, or 16. To calculate the probability, we can use the following formula: CONCEPT Two-Way Tables/Contingency Tables Report an issue with this question 9

A basketball player makes 60% of his free throws. We set him on the free throw line and asked him to shoot free throws until he misses. Let the random variable X be the number of free throws taken by the player until he misses. Assuming that his shots are independent, find the probability that he will miss the shot on his 6th throw.  0.01866  0.03110  0.04666  0.00614 RATIONALE Since we are looking for the probability until the first success, we will use the following Geometric distribution formula: The variable k is the number of trials until the first success, which in this case, is 6 throws. The variable p is the probability of success, which in this case, a success is considered missing a free throw. If the basketball player has a 60% of making it, he has a 40%, or 0.40, chance of missing. CONCEPT Geometric Distribution Report an issue with this question 10 Using this Venn diagram, what is the probability that event A or event B occurs?  0.41

 0.68  0.36  0.77 RATIONALE To find the probability that event A or event B occurs, we can use the following formula for overlapping events: The probability of event A is ALL of circle A, or 0.24 + 0.41 = 0.65. The probability of event B is ALL of circle B, or 0.12 + 0.41 = 0.53. The probability of event A and B is the intersection of the Venn diagram, or 0.41. We can also simply add up all the parts = 0.24 + 0.41 + 0.12 = 0.77. CONCEPT "Either/Or" Probability for Overlapping Events Report an issue with this question 11 Carl throws a single die twice in a row. For the first throw, Carl rolled a 2; for the second throw he rolled a 4. What is the probability of rolling a 2 and then a 4? Answer choices are in the form of a percentage, rounded to the nearest whole number.  36%  33%  3%  22% RATIONALE

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The two events (first roll and second roll) are independent of each other. To find the probability of getting a 2 on the first roll and getting a 4 on the second roll, we can use the following formula: Note that the probability of rolling a two is , and rolling a four is the same probability, . CONCEPT "And" Probability for Independent Events Report an issue with this question 12 John randomly selects a ball from a bag containing different colored balls. The odds in favor of his picking a red ball are 3:11. What is the probability ratio for John picking a red ball from the bag?     RATIONALE Recall that we can go from " " odds to a probability by rewriting it as the fraction " ". So odds of 3:11 is equivalent to the following probability:

CONCEPT Odds Report an issue with this question 13 Paul went to a blackjack table at the casino. At the table, the dealer has just shuffled a standard deck of 52 cards. Paul has had good luck at blackjack in the past, and he actually got three blackjacks with Kings in a row the last time he played. Because of this lucky run, Paul thinks that Kings are the luckiest card. The dealer deals the first card to him. In a split second, he can see that it is a black card, but he is unsure if it is a King. What is the probability of the card being a King, given that it is a black card? Answer choices are in a percentage format, rounded to the nearest whole number.  67%  8%  23%  50% RATIONALE The probability of it being a King given it is a Black card uses the conditional formula: Note that in a standard deck of 52 cards, half of the cards are black, or 26 out of 52. Of those 26 black cards, only two are Kings. CONCEPT Conditional Probability

Report an issue with this question 14 Tracie spins the four-colored spinner shown below. She records the total number of times the spinner lands on the color red and constructs a graph to visualize her results. Which of the following statements is TRUE?  If Tracie spins the spinner 4 times, it will land on red at least once.  If Tracie spins the spinner 1,000 times, it would land on red close to 250 times.  The theoretical probability of the spinner landing on red will change with every spin completed.  If Tracie spins the spinner 1,000 times, the relative frequency of it landing on red will remain constant. RATIONALE If we make the assumption that the area of the colors represents the true proportion, then each color is equally weighted. Since there are four colors we would expect them to come up roughly 1/4 of the time. So on 1000 rolls the expected value = n*p = 1000*0.25 = 250. CONCEPT Law of Large Numbers/Law of Averages Report an issue with this question 15 Sadie is selecting two pieces of paper at random from the stack of colored paper in her closet. The stack contains several sheets of each of the standard colors: red, orange, yellow, green, blue, and violet. All of the following are possible outcomes for Sadie's selection, EXCEPT:  Red, red  Orange, yellow  Blue, black

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 Green, violet RATIONALE Since black is not part of the original set, it cannot be chosen into the sample. CONCEPT Outcomes and Events Report an issue with this question 16 Two sets A and B are shown in the Venn diagram below. Which statement is TRUE?  Set B has 5 elements.  There are a total of 17 elements shown in the Venn diagram.  Sets A and B have 15 common elements.  Set A has 12 elements. RATIONALE The number of elements of Set A is everything in Circle A, or 10+2 = 12 elements. The number of elements of Set B is everything in Circle B, or 5+2 = 7 elements, not 5 elements. The intersection, or middle section, would show the common elements, which is 2 elements, not 15 elements. To get the total number of items in the Venn diagram, we add up what is in A and B and outside, which is 10+2+5+3 = 20 elements, not 17 elements. CONCEPT Venn Diagrams Report an issue with this question

17 Which of the following situations describes a continuous distribution?  A probability distribution showing the number of pages employees read during the workday.  A probability distribution showing the number of minutes employees spend at lunch.  A probability distribution of the average time it takes employees to drive to work.  A probability distribution of the workers who arrive late to work each day. RATIONALE For a distribution to be continuous, there must be an infinite number of possibilities. Since we are measuring the time to drive to work, there are an infinite number of values we might observe, for example: 2 hours, 30 minutes, 40 seconds, etc. CONCEPT Probability Distribution Report an issue with this question 18 Which of the following is a property of binomial distributions?  All trials are dependent.  The sum of the probabilities of successes and failures is always 1.  The expected value is equal to the number of successes in the experiment.  There are exactly three possible outcomes for each trial. RATIONALE Recall that for any probability distribution, the sum of all the probabilities must sum to 1. CONCEPT

Binomial Distribution Report an issue with this question 19 John is playing a game with a standard deck of playing cards. He wants to draw a jack on the first try. Which of the following statements is true?  The probability that John draws a jack on the first try is 3/13. If John replaces the card, re-shuffles, and draws again, the probability that he will pull another jack stays the same.  The probability that John draws a jack on the first try is 1/13. If John replaces the card, re-shuffles, and draws again, the probability that he will pull another jack stays the same.  The probability that John draws a jack on the first try is 1/13. If John replaces the card, re-shuffles, and draws again, the probability that he will pull another jack increases.  The probability that John draws a jack on the first try is 1/13. If John replaces the card, re-shuffles, and draws again, the probability that he will pull another jack decreases. RATIONALE Events are said to be independent if one event does not influence the likelihood of the other. Since John re-shuffles the deck and puts the card back in the deck, the probability should be the same and the first draw will not influence the second. CONCEPT Independent vs. Dependent Events Report an issue with this question 20 Dan is playing a game where he selects a card from a deck of four cards, labeled 1, 2, 3, or 4. The possible cards and probabilities are shown in the probability distribution below.

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What is the expected value for the card that Dan selects?  3.5  2.0  2.5  1.0 RATIONALE The expected value, also called the mean of a probability distribution, is found by adding the products of each individual outcome and its probability. We can use the following formula to calculate the expected value, E(X): CONCEPT Expected Value Report an issue with this question 21 Which of the following is an example of a false positive?  Test results indicate that a patient does not have cancer when, in fact, he does.  Test results indicate that a patient has cancer when, in fact, he does not.  Test results confirm that a patient does not have cancer.  Test results confirm that a patient has cancer. RATIONALE Since the test results indicate positively that the patient has cancer, when in fact cancer is not present, this is a false positive.

CONCEPT False Positives/False Negatives Report an issue with this question 22 The gender and age of Acme Painting Company's employees are shown below. Age Gender 23 Female 23 Male 24 Female 26 Female 27 Male 28 Male 30 Male 31 Female 33 Male 33 Female 33 Female 34 Male 36 Male 37 Male 38 Female 40 Female 42 Male 44 Female If the CEO is selecting one employee at random, what is the chance he will select a male OR someone in their 40s?  1/2  11/18

 1/18  1/3 RATIONALE Since it is possible for an employee to be a male and a person in their 40s, these two events are overlapping. We can use the following formula: Of the 18 employees, there are 9 females and 9 males, so . There are a total of 3 people in their 40s, so . Of the people in their 40s, only one is male so . CONCEPT "Either/Or" Probability for Overlapping Events Report an issue with this question 23 Zhi and her friends moved on to the card tables at the casino. Zhi wanted to figure out the probability of drawing a King of clubs or an Ace of clubs. Choose the correct probability of drawing a King of clubs or an Ace of clubs. Answer choices are in the form of a percentage, rounded to the nearest whole number.  2%  4%  6%  8%

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RATIONALE Since the two events, drawing a King of Clubs and drawing an Ace of Clubs, are non-overlapping, we can use the following formula: CONCEPT "Either/Or" Probability for Non-Overlapping Events Report an issue with this question 24 Fifty people were asked whether they were left handed. Six people answered "yes." What is the relative frequency of left-handed people in this group? Answer choices are rounded to the hundredths place.  8.33  1.14  0.88  0.12 RATIONALE The relative frequency of a left hand is: CONCEPT Relative Frequency Probability/Empirical Method Report an issue with this question 25 What is the theoretical probability of drawing a king from a well shuffled deck of 52 cards?

    RATIONALE Recall that there are four kings in a standard deck of cards. The probability of a king is: CONCEPT Theoretical Probability/A Priori Method Report an issue with this question 26 The average number of tunnel construction projects that take place at any one time in a certain state is 3. Find the probability of exactly five tunnel construction projects taking place in this state.  0.10  0.023  0.020  0.048 RATIONALE

Since we are finding the probability of a given number of events happening in a fixed interval when the events occur independently and the average rate of occurrence is known, we can use the following Poisson distribution formula: The variable k is the given number of occurrences, which in this case, is 5 projects. The variable λ is the average rate of event occurrences, which in this case, is 3 projects. CONCEPT Poisson Distribution Report an issue with this question 27 Select the following statement that describes overlapping events.  Amanda understands that she cannot get a black diamond when playing poker.  Amanda wants a black card so she can have a winning hand, and she receives the two of hearts.  Receiving a Jack of diamonds meets the requirement of getting both a Jack and a diamond.  Amanda rolls a three when she needed to roll an even number. RATIONALE Events are overlapping if the two events can both occur in a single trial of a chance experiment. Since she wants a Jack {Jack of Hearts, Jack of Clubs, Jack of Diamonds, Jack of Spades} and a diamond {Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, or King: all as diamonds}, the overlap is Jack of Diamonds.

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CONCEPT Overlapping Events Report an issue with this question  About  Contact Us  Privacy Policy  Cookie Policy  Terms of Use  Your Privacy Choices © 2023 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC.

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Sophia Milestone 3 exam

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