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ENGLISH 215 Statistics Milestone3Kita.docx La'Vonn rolled a die 100 times. His results are below. What is the relative frequency for La'Vonn rolling a 3? Answer choices are r... ounded to the hundredths place. 2 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. 3 For a math assignment, Michelle rolls a set of three standard dice at the same time and notes the results of each trial. What is the total number of outcomes for each trial? 4 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. 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 5 Which of the following is an example of a false positive? • Test results confirm that a patient does not have cancer. • Test results indicate that a patient has cancer when, in fact, he does not. • Test results confirm that a patient has cancer. • Test results indicate that a patient does not have cancer when, in fact, he does. 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 6 Select the following statement that describes overlapping events. 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. CONCEPT Overlapping Events 7 Mark noticed that the probability that a certain player hits a home run in a single game is 0.165. Mark is interested in the variability of the number of home runs if this player plays 150 games. If Mark uses the normal approximation of the binomial distribution to model the number of home runs, what is the variance for a total of 150 games? Answer choices are rounded to the hundredths place. RATIONALE In this situation, we know: n = sample size = 150 p = success probability = 0.165 We can also say that q, or the complement of p, equals: q = 1 - p = 1 - 0.165 = 0.835 The variance is equivalent to n*p*q: CONCEPT Normal Distribution Approximation of the Binomial Distribution 8 Two sets A and B are shown in the Venn diagram below. Which statement is TRUE? 8 3 7 2 • Sets A and B have 3 common elements. • Set A has 8 elements. • Set B has 7 elements. • There are a total of 2 elements shown in the Venn diagram. RATIONALE The intersection, or middle section, would show the common elements, which is 3. The number of elements of Set A is everything in Circle A, or 8+3 = 11 elements, not 8 elements. The number of elements of Set B is everything in Circle B, or 7+3 = 10 elements, not 7 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 8+3+7+2=20 elements, not 2 elements. CONCEPT Venn Diagrams 9 Jake tosses a coin and rolls a six-sided die. All of the following are possible outcomes EXCEPT: RATIONALE Recall a coin has heads and tails and a standard die has six values, {1, 2, 3, 4, 5, or 6}. So, obtaining a value of 7 is not possible. CONCEPT Outcomes and Events 10 The average number of babies born at a private hospital's maternity wing is 6 per hour. What is the probability that three babies are born during a particular 1-hour period in this maternity wing? • 0.09 • 0.16 • 0.20 • 0.13 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 3 babies. The variable λ is the average rate of event occurrences, which in this case, is 6 babies. CONCEPT Poisson Distribution 11 Kendra was trying to decide which type of frozen yogurt to restock based on popularity: flavors with chocolate or flavors without chocolate. After studying the data, she noticed that chocolate flavors sold best on the weekdays and on the weekends, but not best overall. Which paradox has Kendra encountered? • 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 chocolate flavors is larger on the weekends, but this trend changes when looking at sales overall, this is a reversal of the trend. CONCEPT Paradoxes 12 What is the probability of NOT drawing a Queen from a standard deck of 52 cards? 12/13 • • • • 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 there are a total of 4 Queen cards in a standard deck of 52 cards. So the probability of NOT getting a Queen is equivalent to: CONCEPT Complement of an Event 13 Using the Venn Diagram below, what is the conditional probability of event B occurring, assuming event A has happened [P(B|A)]? 0.24 0.41 0.12 0.23 • 0.41 • 0.63 • 0.24 • 0.77 RATIONALE To get the probability of B given A 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 0.41. The probability of A is all of Circle A, or 0.24 + 0.41 = 0.65. CONCEPT Conditional Probability 14 Eric is randomly drawing cards from a deck of 52. He first draws a red card, places it back in the deck, shuffles the deck, and then draws another card. What is the probability of drawing a red card, placing it back in the deck, and drawing another red card? Answer choices are in the form of a percentage, rounded to the nearest whole number. 4% • 13% • 25% RATIONALE Since Eric puts the card back and re-shuffles, the two events (first draw and second draw) are independent of each other. To find the probability of red on the first draw and second draw, we can use the following formula: Note that the probability of drawing a red card is or for each event. CONCEPT "And" Probability for Independent Events 15 Sarah throws a fair die multiple times, recording the total number of "2"s she throws and then calculating the proportion of "2"s she has thrown so far after each throw. She then constructs a graph to visualize her results. Which of the following statements is FALSE? RATIONALE The probability distribution for the outcomes doesn't change; however, the sampling distribution for the outcomes does. CONCEPT Law of Large Numbers/Law of Averages 16 Satara was having fun playing poker. She needed the next two cards dealt to be hearts so she could make a flush (five cards of the same suit). There are 10 cards left in the deck, and three are hearts. What is the probability that the two cards dealt to Satara (without replacement) will both be hearts? Answer choices are in percentage format, rounded to the nearest whole number. • 7% • 60% • 26% • 30% RATIONALE If there are 10 cards left in the deck with 3 hearts, the probability of being dealt 2 hearts 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 heart would be 3 out of 10, or . The probability that the second card is a heart, given that the first card was also a heart, would be because we now have only 9 cards remaining and only two of those cards are hearts (since the first card was a heart). So we can use these probabilities to find the probability that the two cards will both be hearts: CONCEPT "And" Probability for Dependent Events 17 Using this Venn diagram, what is the probability that event A or event B occurs? 0.24 0.41 0.12 0.23 • 0.68 • 0.41 • 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 18 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? 3/14 • • • • 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 19 Which of the following is a property of binomial distributions? RATIONALE Recall that for any probability distribution, the sum of all the probabilities must sum to 1. CONCEPT Binomial Distribution 20 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. • 6% • 2% • 4% • 8% 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 21 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. What is the expected value for the card that Dan selects? 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 22 Which of the following situations describes a continuous distribution? RATIONALE Since the weight of newborns can be an infinite number of values, such as 8 pounds, 9 ounces, etc, this would be an example of a continuous distribution. CONCEPT Probability Distribution 23 A survey asked 1,000 people which magazine they preferred, given three choices. The table below breaks the votes down by magazine and age group. If you randomly select a person under the age of 40 from the group, what is the probability that they voted for "The Month?" Answer choices are rounded to the hundredths place. RATIONALE The probability of a person voting for "The Month" given he or she is younger than 40 is a conditional probability. We can use the following formula: Remember, to find the total number of Younger than 40, we need to add all values in this column: 104 + 120 + 240 = 464. CONCEPT Conditional Probability and Contingency Tables 24 What is the theoretical probability of drawing a king from a well shuffled deck of 52 cards? 1/13 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 25 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? 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 26 A credit card company surveys 125 of their customers to ask about satisfaction with customer service. The results of the survey, divided by gender, are shown below. 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. 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 27 What is the probability of drawing a spade or a jack from a standard deck of 52 cards? 4/13 • • • • RATIONALE Since it is possible for a card to be a spade and a jack, these two events are overlapping. We can use the following formula: In a standard deck of cards, there are 13 cards that have Spade as their suit, so . There is a total of 4 Jacks, so . Of the 4 Jacks, only one is spade so . [Show More]

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