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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?
Paradoxes 2
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.
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.
Conditional Probability 3
Which of the following situations describes a continuous distribution?
Probability Distribution 4
Which of the following is a condition of binomial probability distributions?
.
Binomial Distribution 5
Which of the following is an example of a false positive?
False Positives/False Negatives 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.
Relative Frequency Probability/Empirical Method 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?
8
Select the following statement that describes non-overlapping events.
Overlapping Events
9
Using the Venn Diagram below, what is the conditional probability of event Q occurring, assuming that event P has already happened [P (Q|P)]?
0.05
0.75
0.73
0.55
RATIONALE
To get the probability of Q given P has occurred, we can use the following conditional formula:
The probability of Q and P is the intersection, or overlap, of the Venn diagram, which is 0.4. The probability of P is all of Circle P, or 0.15 + 0.4 = 0.55.
CONCEPT
Conditional Probability 10
Dida bought a scratch ticket for $2.00. The potential payoffs and probability of those payoffs are shown below.
Value (in $) 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?
$1.50
$2.00
$0.50
$1.15
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 11
What is the probability of NOT drawing a face card 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 the event happening, and then subtracting that from 1. Note that there are a total of 12 face cards in a standard deck of 52 cards. So the probability of NOT getting a face card is equivalent to:
CONCEPT
Complement of an Event
12
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%
6%
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 13
Two sets A and B are shown in the Venn diagram below.
Which statement is FALSE?
Set A has 12 elements.
Sets A and B have 5 common elements.
There are a total of 25 elements shown in the Venn diagram.
Set B has 10 elements.
RATIONALE
To get the total number of items in the Venn diagram, we add up what is in A and B and outside, which is 7+5+5+3 =20 elements, not 25 elements. The intersection, or middle section, would show the common elements, which is 5 elements.
The number of elements of Set A is everything in Circle A, or 7+5 = 12 elements. The number of elements of Set B is everything in Circle B, or 5+5 = 10 elements.
CONCEPT
Venn Diagrams 14
Using this Venn diagram, what is the probability that event A or event B occurs?
0.78
0.22
0.42
0.60
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 0.39 + 0.18 = 0.57. The probability of event B is ALL of circle B, or 0.21 + 0.18 = 0.39.
The probability of event A and B is the intersection of the Venn diagram, or 0.18. We can also simply add up all the parts = 0.39 + 0.18 + 0.21 = 0.78.
CONCEPT
"Either/Or" Probability for Overlapping Events 15
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?
52
676
26
650
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 16
Phil is randomly drawing cards from a deck of 52. He first draws a Queen, places it back in the deck, shuffles the deck, and then draws another card.
What is the probability of drawing a Queen, placing it back in the deck, and drawing any face card? Answer choices are in the form of a percentage, rounded to the nearest whole number.
31%
2%
7%
25%
RATIONALE
Since Phil 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 getting a Queen on the first draw and a face card on the second draw, we can use the following formula:
CONCEPT
card is .
"And" Probability for Independent Events 17
Tim rolls two six-sided dice and flips a coin.
All of the following are possible outcomes, EXCEPT:
5, 2, Tails
1, Tails, 6
Heads, 3, 4
2, 8, Heads
RATIONALE
Recall that a standard coin has two values, {Heads or Tails}, while a standard die has six values {1, 2, 3, 4, 5, or 6}. So, obtaining a 2 is possible, however the 8 is not.
CONCEPT
Outcomes and Events 18
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?
1/18
1/3
11/18
1/2
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 19
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?
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.
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.
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 20
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.
0.048
0.023
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:
P left parenthesis X equals k right parenthesis equals fraction numerator lambda to the power of k e to the power of negative lambda end exponent over denominator k factorial end fraction
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 21
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.
0.62
0.24
0.13
0.38
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.
CONCEPT
Conditional Probability and Contingency Tables 22
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.
Age Below 40 Age 40 and Above
The National Journal 104 200
Newsday 120 230
The Month 240 106
If a survey is selected at random, what is the probability that the person voted for "Newsday" and is also age 40 or older? Answer choices are rounded to the hundredths place.
0.54
0.23
0.34
0.66
RATIONALE
If we want the probability of people who voted for "Newsday" and are also age 40 and over, we just need to look at the box that is associated with both categories, or 230. To calculate the probability, we can use the following formula:
CONCEPT
Two-Way Tables/Contingency Tables 23
Mark looked at the statistics for his favorite baseball player, Jose Bautista. Mark looked at seasons when Bautista played 100 or more games and found that Bautista's probability of hitting a home run in a game is 0.173.
If Mark uses the normal approximation of the binomial distribution, what will be the variance of the number of home runs Bautista is projected to hit in 100 games? Answer choices are rounded to the tenths place.
17.3
0.8
14.3
3.8
RATIONALE
In this situation, we know:
n = sample size = 100
p = success probability = 0.173
We can also say that q, or the complement of p, equals: q = 1 - p = 1 - 0.173 = 0.827
The variance is equivalent to n*p*q:
CONCEPT
Normal Distribution Approximation of the Binomial Distribution 24
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.
The theoretical probability of the spinner landing on red will change with every spin completed.
If Tracie spins the spinner 1,000 times, it would land on red close to 250 times.
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 25
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:
P left parenthesis X equals k right parenthesis equals left parenthesis 1 minus p right parenthesis to the power of k minus 1 end exponent p
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 26
Patricia was having fun playing poker. She needed the next two cards dealt to be spades so she could make a flush (five cards of the same suit). There are 12 cards left in the deck, and three are spades.
What is the probability that the two cards dealt to Patricia (without replacement) will both be spades? Answer choices are in percentage format, rounded to the nearest whole number.
18%
5%
17%
25%
RATIONALE
If there are 12 cards left in the deck with 3 spades, the probability of being dealt 2 spades 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 spade would be 3 out of 12, or . The probability that the second card is a spade, given that the first card
was also a spade, would be because we now have only 11 cards remaining and only two of those cards are spades (since the first card was a spade).
So we can use these probabilities to find the probability that the two cards will both be spades:
CONCEPT
"And" Probability for Dependent Events 27
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
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