Sophia - Statistics - Unit 3 - Milestone 3

1 CONCEPT → Independent vs. Dependent Events 2 CONCEPT → "And" Probability for Dependent Events 3 CONCEPT → Law of Large Numbers/Law of Averages 4 CONCEPT → "Either/Or" Probability for Non-Overlapping Events 5 CONCEPT → Expected Value 6 CONCEPT → Conditional Probability 7 CONCEPT → "Either/Or" Probability for Overlapping Events 8 CONCEPT → Paradoxes 9 CONCEPT → Conditional Probability 10 CONCEPT → Conditional Probability and Contingency Tables 11 CONCEPT → Fundamental Counting Principle 12 CONCEPT → False Positives/False Negatives 13 CONCEPT → Overlapping Events 14 CONCEPT → Theoretical Probability/A Priori Method 15 25/27 " that's 93% RETAKE # 25 questions were answered correctly . # 2 questions were answered incorrectly . A bag holds 20 red marbles and 40 green ones, for a total of 60 marbles. Ryan picks one marble from the bag at random, hoping to pick a red marble. Which of the following statements is true? RATIONALE The probability that Ryan will pick a red marble on the first try would be 20/60, or 33%. If Ryan keeps the red marble and doesn't replace it, then the likelihood of a green marble goes from 40/60, or 0.66, from the first try, to 40/59, or 0.678, on the second try. The second try has a higher probability. Report an issue with this question 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. 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 a ff ect 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: Report an issue with this question Joseph selects a card from a standard deck of 52 cards and then replaces it afterwards. He decides to record the total number of red cards that he selects and calculates the proportion of red cards that he has selected so far after each pick. He then constructs a graph to visualize his results. Which of the following statements is FALSE? RATIONALE Since we are only graphing the proportion of red cards, there is no information in the sampling distribution about the actual suit of the card. Report an issue with this question 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. 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: Report an issue with this question Dida bought a scratch ticket for $2.00. The potential payo ff s and probability of those payo ff s are shown below. Value Probability $0.00 0.15 $0.50 0.50 $1.00 0.20 $2.00 0.10 $10.00 0.05 What is the expected value for the scratch ticket that Dida bought? 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): Report an issue with this question Luke went to a blackjack table at the casino. At the table, the dealer has just shu ffl ed a standard deck of 52 cards. Luke has had good luck at blackjack in the past, and he actually got three blackjacks with Queens in a row the last time he played. Because of this lucky run, Luke thinks that Queens are the luckiest card. The dealer deals the first card to him. In a split second, he can see that it is a face card, but he is unsure if it is a Queen. What is the probability of the card being a Queen, given that it is a face card? Answer choices are in a percentage format, rounded to the nearest whole number. RATIONALE The probability of it being a Queen given it is a Face card uses the conditional formula: Note that there are 12 out of 52 that are face cards. Of those 12 cards, only 4 of them are also Queens. Report an issue with this question Using this Venn diagram, what is the probability that event A or event B occurs? 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 0.43 + 0.11 = 0.54. The probability of event B is ALL of circle B, or 0.23 + 0.11 = 0.34. The probability of event A and B is the intersection of the Venn diagram, or 0.11. We can also simply add up all the parts = 0.43 + 0.11 + 0.23 = 0.77. Report an issue with this question 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? 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. Report an issue with this question Using the Venn Diagram below, what is the conditional probability of event A occurring, assuming that event B has already occurred [P(A|B)]? 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. Report an issue with this question 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 Females Extremely Satisfied 25 7 Satisfied 21 13 Neutral 13 16 Dissatisfied 9 14 Extremely Dissatisfied 2 5 If you were to choose a female from the group, what is the probability that she is satisfied with the company's customer service? Answer choices are rounded to the hundredths place. RATIONALE The probability of a person being "satisfied" given she is a female is a conditional probability. We can use the following formula: Remember, to find the total number of females, we need to add all values in this column: 7 + 13 + 16 + 14 + 5 = 55. Report an issue with this question For a math assignment, Jane has to roll a set of six standard dice and record the results of each trial. She wonders how many di ff erent outcomes are possible after rolling all six dice. What is the total number of possible outcomes for each trial? 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. Each die has 6 possible outcomes, so the overall number of outcomes for rolling 6 die with 6 possible outcomes each is: Report an issue with this question Which of the following is an example of a false positive? RATIONALE Sending a man to jail, when in fact he is innocent, is a false positive. Report an issue with this question 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. Report an issue with this question What is the theoretical probability of drawing a king from a well shu ffl ed deck of 52 cards? RATIONALE Recall that there are four kings in a standard deck of cards. The probability of a king is: Report an issue with this question Maria flipped a coin 60 times, and the coin came up tails 32 times. What is the relative frequency of the coin turning up heads in this experiment? Answer choices are rounded to the hundredths place. ● " # " ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # # ● # ● The probability that Ryan will pick a red marble on the first try is 33%. If he keeps this marble and picks another from the bag, the probability that he will pick a green marble decreases. ● The probability that Ryan will pick a red marble on the first try is 33%. If he keeps this marble and picks another from the bag, the probability that he will pick a green marble increases. $ The probability that Ryan will pick a red marble on the first try is 67%. If he keeps this marble and picks another from the bag, the probability that he will pick a green marble decreases. ● The probability that Ryan will pick a red marble on the first try is 67%. If he keeps this marble and picks another from the bag, the probability that he will pick a green marble increases. ● 33% ● 29% $ 10% ● 13% ● The theoretical probability for selecting a red card in each pick is 0.5. ● The relative frequency of selecting a red card changes with the increase in number of selections. ● This is an example of the law of large numbers. $ For a given number of selections, we can use Joseph's graph to find the number of diamonds that he has picked so far. ● 8% ● 2% ● 6% $ 4% ● $0.50 $ $1.15 ● $2.00 ● $1.50 ● 8% ● 77% ● 4% $ 33% ● 0.66 $ 0.77 ● 0.23 ● 0.11 ● False Negative ● Benford's Law $ Simpson's Paradox ● False Positive ● 0.05 ● 0.71 $ 0.22 ● 0.10 ● 0.13 $ 0.24 ● 0.38 ● 0.62 ● 7,776 ● 36 $ 46,656 ● 216 ● Sending a guilty man to jail. $ Sending an innocent man to jail. ● A medical test coming back positive for a disease you do have. ● A medical test coming back negative for a disease you don't have. ● 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. ● Amanda understands that she cannot get a black diamond when playing poker. $ ● ● ● ● 0.53 UNIT 3 — MILESTONE 3 SCORE 25/27