Question 1

If one card is randomly picked from a standard deck of 52 cards, the probability that the card will be a red suit (Heart or Diamond), or a face card (Jack, Queen, or King), or both, is
A) 50.0% (26/52)
B) 61.5% (32/52)
C) 76.9% (40/52)
D) 88.5% (46/52)

Accepted Answer

To calculate the probability, we consider the total number of red cards (26) plus the number of black face cards (6 for Jacks, Queens, and Kings in Spades and Clubs), which gives us 32 unique cards that fit the criteria out of 52.

Question 2

Use the following information for questions:
 A standard 52-card deck is shuffled and 2 cards are picked from the top of the deck.
-The probability that the first card is a Heart and the second card is a Spade is
A) 5.9%
B) 6.3%
C) 6.4%
D) 25.0%

Accepted Answer

The probability of drawing a Heart first is 13/52 (since there are 13 Hearts in a deck of 52 cards). After drawing a Heart, there are 51 cards left, with 13 of them being Spades. So, the probability of then drawing a Spade is 13/51. Multiplying these probabilities gives (13/52) * (13/51) = 169/2652 ≈ 6.4%.

Question 3

Use the following information for questions:
 A standard 52-card deck is shuffled and 2 cards are picked from the top of the deck.
-The probability that the both cards are Hearts is
A) 6.3%
B) 5.9%
C) 6.4%
D) 25.0%

Accepted Answer

The probability of the first card being a Heart is 13/52. If the first card is a Heart, the probability of the second card being a Heart is 12/51. Thus, the probability of both cards being Hearts is (13/52) * (12/51) = 1/17 ≈ 5.9%.

Question 4

Use the following information for questions:
 A standard 52-card deck is shuffled and 2 cards are picked from the top of the deck.
-The probability that the first card is a face card (Jack, Queen, King) and the second card is not a face card is
A) 5.3%
B) 17.8%
C) 18.1%
D) 59.2%

Accepted Answer

There are 12 face cards in a 52-card deck. The probability of drawing a face card first is 12/52. If a face card is drawn first, there are 51 cards left, with 40 being non-face cards. The probability of drawing a non-face card second is 40/51. The combined probability is (12/52) * (40/51) = 18.1%.

Question 5

Two standard 52-card decks are shuffled and two cards are picked at random - one card from each deck. The probability that two Hearts are drawn is
A) 5.9%
B) 6.3%
C) 25.0 %
D) 50.0%

Accepted Answer

Each deck has 13 Hearts out of 52 cards. The probability of drawing a Heart from one deck is 13/52. Since the decks are independent, the probability of drawing a Heart from both decks is (13/52) * (13/52) = 1/16 = 6.25%, which is approximately 6.3%.

Question 6

Use the following information for questions:
 A card is drawn at random from a standard 52-card deck.
-The conditional probability that the card is a King given that a face card (Jack, Queen, or King) was drawn is
A) 20.0% (1/5)
B) 25.0% (1/4)
C) 33.3% (1/3)
D) 50.0% (1/2)

Accepted Answer

There are 12 face cards in a deck (4 Jacks, 4 Queens, 4 Kings). Given that a face card is drawn, the probability it is a King is 4 out of 12, which simplifies to 1/3 or 33.3%.

Question 7

Use the following information for questions:
 A card is drawn at random from a standard 52-card deck.
-The conditional probability that the card is a 2 given that a 2 or a 3 was drawn is
A) 20.0% (1/5)
B) 25.0% (1/4)
C) 33.3% (1/3)
D) 50.0% (1/2)

Accepted Answer

Given that a 2 or a 3 was drawn, there are 8 possible outcomes (4 twos and 4 threes). The probability that the card is a 2 out of these 8 possible outcomes is 4/8, which simplifies to 1/2 or 50%.

Question 8

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.2 and winning is independent from one bottle to the next. What is the probability that a customer must open three or more bottles before winning a prize?
A) (0.2)(0.2)(0.8) = 0.032
B) (0.8)(0.8)(0.2) = 0.128
C) (0.8)(0.8) = 0.64
D) 1 - (0.2)(0.2)(0.8) = .968

Accepted Answer

The probability of not winning a prize on the first and second tries is (0.8)(0.8) = 0.64, which means the customer must open three or more bottles before winning.

Question 9

Use the following information for questions:
 Suppose two different states each pick a two-digit lottery number between 00 and 99 (for a 100 possible numbers).
-What is the probability that both states pick the number 13?
A) 2/100
B) 1/100
C) 1/200
D) 1/10,000

Accepted Answer

The probability of one state picking the number 13 is 1/100. Since the events are independent, the probability that both states pick the number 13 is (1/100) * (1/100) = 1/10,000.

Question 10

Use the following information for questions:
 Suppose two different states each pick a two-digit lottery number between 00 and 99 (for a 100 possible numbers).
-What is the probability that both states pick the same number?
A) 2/100
B) 1/100
C) 1/200
D) 1/10,000

Accepted Answer

The probability that both states pick the same number is 1/100 because the first state can pick any number (100 options), and the second state has a 1/100 chance of picking the same number as the first state.

Question 11

A six-sided die is made that has four Green sides and two Red sides, all equally likely to land face up when the die is tossed. The die is tossed three times. Which of these sequences (in the order shown) has the highest probability?
A) Green, Green, Green
B) Green, Green, Red
C) Green, Red, Red
D) They are all equally likely

Accepted Answer

The probability of each sequence is calculated by multiplying the probabilities of each individual outcome. Sequence A has the highest probability because it involves more Green outcomes, which are more likely (4/6) compared to Red outcomes (2/6).

Question 12

Use the following information for questions:
 A student doing an internship at a large research firm collected the following data, representing all of the studies the firm had conducted over the past 3 years. 
-Suppose one study is to be randomly selected from all studies conducted over the past 3 years by this large research firm. What is the probability that this study used randomization?
A) 0.1167
B) 0.2333
C) 0.26
D) 0.3111

Accepted Answer

The probability that a study used randomization is calculated by dividing the number of studies that used randomization by the total number of studies. The given probability of 0.2333 suggests that this is the correct proportion.

Question 13

Use the following information for questions:
 A student doing an internship at a large research firm collected the following data, representing all of the studies the firm had conducted over the past 3 years. 
-What is the probability that the study used randomization, if we already know that the study was a retrospective observational study?
A) 0.1167
B) 0.2333
C) 0.26
D) 0.3111

Accepted Answer

Retrospective observational studies typically do not use randomization, as they analyze existing data collected for other purposes. Therefore, the probability is very low, closest to 0.1167.

Question 14

Use the following information for questions:
 A student doing an internship at a large research firm collected the following data, representing all of the studies the firm had conducted over the past 3 years. 
-Define the events E = {the study was an experiment}, U = {the study used randomization} and R = {the study was a retrospective observational study}. Which of the following sets of events are disjoint?
A) E and U
B) E and R
C) U and R
D) All three
E) None of the above

Accepted Answer

E and R are disjoint because an experiment (E) involves manipulating variables to determine effects, while a retrospective observational study (R) involves looking back at data without manipulation of variables.

Question 15

Use the following information for questions:
 A student doing an internship at a large research firm collected the following data, representing all of the studies the firm had conducted over the past 3 years. 
-Define the events E = {the study was an experiment} and U = {the study used randomization}. Are the events E and U independent?
A) Yes
B) No
C) Can't tell
D) Need more information to determine this.

Accepted Answer

The events E and U are independent because the occurrence of one event does not affect the probability of the other event occurring.

Question 16

Use the following information for questions:
 For each of the following statements, determine if they are true or false.
-The probability of the intersection of two events A and B, and the probability of the union of A and B can never be equal.

Accepted Answer

The probability of the intersection of two events A and B can be equal to the probability of the union of A and B if A and B are the same event or if A is a subset of B (or vice versa), making their intersection and union effectively the same set, thus having the same probability.

Question 17

Use the following information for questions:
 For each of the following statements, determine if they are true or false.
-If events A and B are known to be independent and P(A) = 0.2 and P(B) = 0.3, then P(A and B) = 0.5.

Accepted Answer

Since events A and B are independent, P(A and B) = P(A) * P(B) = 0.2 * 0.3 = 0.06, not 0.5.

Question 18

Use the following information for questions:
 For each of the following statements, determine if they are true or false.
-If events D and E are known to be mutually exclusive, then P(D|E) = 0.

Accepted Answer

If events D and E are mutually exclusive, it means they cannot both occur at the same time, hence the probability of D occurring given that E has occurred is 0.

Question 19

Which of the following is definitely true for mutually exclusive events A and B?
A) P(A) = 1 - P(B)
B) P(A) + P(B) = 1
C) P(A and B) = P(A)P(B)
D) P(A and B) = 0

Accepted Answer

Mutually exclusive events cannot happen at the same time, which means the probability of A and B happening together is 0.

Question 20

Suppose that the probability of event A is 0.2 and the probability of event B is 0.4. Also, suppose that the two events are independent. Then P(A|B) is:
A) P(A) = 0.2
B) P(A) / P(B) = 0.2/0.4 = ½
C) P(A)P(B) = (0.2)(0.4) = 0.08
D) None of the above.

Accepted Answer

For independent events, the probability of event A given event B, P(A|B), is simply the probability of A, which is 0.2. This is because the occurrence of B does not affect the probability of A in the case of independent events.

Quiz 7: Probability

If one card is randomly picked from a standard deck of 52 cards, the probability that the card will be a red suit (Heart or Diamond) , or a face card (Jack, Queen, or King) , or both, is

Use the following information for questions: A standard 52-card deck is shuffled and 2 cards are picked from the top of the deck. -The probability that the first card is a Heart and the second card is a Spade is

Use the following information for questions: A standard 52-card deck is shuffled and 2 cards are picked from the top of the deck. -The probability that the both cards are Hearts is

Use the following information for questions: A standard 52-card deck is shuffled and 2 cards are picked from the top of the deck. -The probability that the first card is a face card (Jack, Queen, King) and the second card is not a face card is

Two standard 52-card decks are shuffled and two cards are picked at random - one card from each deck. The probability that two Hearts are drawn is

Use the following information for questions: A card is drawn at random from a standard 52-card deck. -The conditional probability that the card is a King given that a face card (Jack, Queen, or King) was drawn is

Use the following information for questions: A card is drawn at random from a standard 52-card deck. -The conditional probability that the card is a 2 given that a 2 or a 3 was drawn is

Use the following information for questions: Suppose two different states each pick a two-digit lottery number between 00 and 99 (for a 100 possible numbers) . -What is the probability that both states pick the number 13?

Use the following information for questions: Suppose two different states each pick a two-digit lottery number between 00 and 99 (for a 100 possible numbers) . -What is the probability that both states pick the same number?

A six-sided die is made that has four Green sides and two Red sides, all equally likely to land face up when the die is tossed. The die is tossed three times. Which of these sequences (in the order shown) has the highest probability?

Use the following information for questions: For each of the following statements, determine if they are true or false. -The probability of the intersection of two events A and B, and the probability of the union of A and B can never be equal.

Use the following information for questions: For each of the following statements, determine if they are true or false. -If events A and B are known to be independent and P(A) = 0.2 and P(B) = 0.3, then P(A and B) = 0.5.

Use the following information for questions: For each of the following statements, determine if they are true or false. -If events D and E are known to be mutually exclusive, then P(D|E) = 0.

Which of the following is definitely true for mutually exclusive events A and B?

Suppose that the probability of event A is 0.2 and the probability of event B is 0.4. Also, suppose that the two events are independent. Then P(A|B) is: