Unit 3- Milestone 3 Statistics

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Jan 9, 2024

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25/27 that's 93% RETAKE 25 questions were answered correctly . 2 questions were answered incorrectly . 1 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?  2.5  3.5  1.0  2.0 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 2

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.  0.023  0.048  0.020  0.10 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 3 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.  25%  4%  22%  13% 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 Report an issue with this question 4 Luke went to a blackjack table at the casino. At the table, the dealer has just shuffled 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.

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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.  77%  8%  4%  33% 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. CONCEPT Conditional Probability Report an issue with this question 5 Which of the following is an example of a false positive?  Test results indicate that a patient has cancer when, in fact, he does not.  Test results indicate that a patient does not have cancer when, in fact, he does.  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 6 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.  1.14  8.33  0.12  0.88 RATIONALE The relative frequency of a left hand is: CONCEPT Relative Frequency Probability/Empirical Method Report an issue with this question 7 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 8 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?  36  27  216  18

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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 3 die with 6 possible outcomes each is: CONCEPT Fundamental Counting Principle Report an issue with this question 9 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 10

Colleen has 6 eggs, one of which is hard-boiled while the rest are raw. Colleen can't remember which of the eggs are raw. Which of the following statements is true?  The probability of Colleen selecting the hard-boiled egg on her first try is 1/5.  The probability of Colleen selecting a raw egg on her first try is 1/6.  If Colleen selected one egg, cracked it open and found out it was raw, the probability of selecting the hard-boiled egg on her second pick is 1/6.  If Colleen selected one egg, cracked it open and found out it was raw, the probability of selecting the hard-boiled egg on her second pick is 1/5. RATIONALE The probability of choosing the hard-boiled egg is 1/6. If she cracks an egg and it is not the hard-boiled egg, then it becomes 1/5 on the next try because there are now only 5 eggs remaining and one has to be the hard-boiled egg as she did not pick it on the first try. CONCEPT Independent vs. Dependent Events Report an issue with this question 11 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.  0.5  0.13  0.19  0.29  0.81 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 12 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? 

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This is an example of the law of large numbers.  The probability distribution for the possible number of outcomes changes as the total number of throws increases.  The relative frequency of "2"s thrown changes as Sarah throws the die more.  The theoretical probability of getting a 2 is 0.167 for each throw. 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 Report an issue with this question 13 A basketball player makes 80% of his free throws. We set him on the free throw line and told 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, what is the probability of this player missing a free throw for the first time on the fifth attempt?  0.4096  0.00128  0.08192  0.0016

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 5 attempts. 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 an 80% of making it, he has a 20%, or 0.20, chance of missing. CONCEPT Geometric Distribution Report an issue with this question 14 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.  0.14  5.37  28.88  5.92 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 15 Using this Venn diagram, what is the probability that event A or event B occurs?  0.36  0.41  0.77  0.68 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.

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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 16 Which of the following is a property of binomial distributions?  There are exactly three possible outcomes for each trial.  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.  All trials are dependent. 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 17 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 G 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.  0.25  0.75  0.13  0.39 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 18 Kyle was trying to decide which type of soda to restock based on popularity: regular cola or diet cola. After studying the data, he noticed that he sold less diet cola on weekdays and weekends. However, after combing through his entire sales records, he actually sold more diet cola than regular cola. Which paradox had Kyle encountered?  Simpson's Paradox  False Negative

 Benford's Law  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 diet coke overall is larger but this trend changes when looking at weekend/weekday, this is a reversal of the trend. CONCEPT Paradoxes Report an issue with this question 19 Jake tosses a coin and rolls a six-sided die. All of the following are possible outcomes EXCEPT:  Tails, One  Heads, Seven  Heads, Five  Tails, Three 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 Report an issue with this question 20

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Select the following statement that describes overlapping events.  Amanda rolls a three when she needed to roll an even number.  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. 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 Report an issue with this question 21 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% 

30%  60%  26% 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 Report an issue with this question 22 Which of the following situations describes a continuous distribution?  A probability distribution showing the number of minutes employees spend at lunch.  A probability distribution showing the number of pages employees read during the workday.

 A probability distribution of the workers who arrive late to work each day.  A probability distribution of the average time it takes employees to drive to work. 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 23 Using the Venn Diagram below, what is the conditional probability of event A occurring, assuming that event B has already occurred [P(A| B)]?  0.22  0.10  0.05  0.71 RATIONALE To get the probability of A given B has occurred, we can use the following conditional formula:

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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 24 Two sets A and B are shown in the Venn diagram below. Which statement is TRUE?  Set A has 12 elements.  Sets A and B have 15 common elements.  There are a total of 17 elements shown in the Venn diagram.  Set B has 5 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 25

What is the probability of NOT drawing a Queen from a standard deck of 52 cards?     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 Report an issue with this question 26 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.

 8%  2%  4%  6% 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 27 What is the probability of drawing a spade or a jack from a standard deck of 52 cards?    

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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 . CONCEPT "Either/Or" Probability for Overlapping Events

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Unit 3- Milestone 3 Statistics

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