Question 1

Pick 5 cards from a randomly shuffled standard deck without replacement; record the number of eights.
A){0,1,2,3,4},not equally likely
B){1,2,3,4,5,6,7,8},equally likely
C){0,1,2,3,4,5},not equally likely
D){1,2,3,4},not equally likely
E){0,1,2,3,4,5,6,7,8},equally likely

Accepted Answer

In a standard deck, there are 4 eights. When picking 5 cards, the number of eights can range from 0 to 4, but these outcomes are not equally likely.

Question 2

Pick cards from a randomly shuffled deck until you get a black card or 4 consecutive red cards; record the cards in order.
A){BBBB,RBBB,RRBB,RRRB,RRRR},not equally likely
B){B,RB,RRB,RRRB,RRRRB,RRRRRB,....},not equally likely
C){B,RB,RRB,RRRB,RRRRB},not equally likely
D){B,RB,RRB,RRRB,RRRR},not equally likely
E){BBBB,BBBR,BBRB,BBRR,BRBB,BRBR,BRRB,BRRR,RBBB,RBBR,RBRB,RBRR,RRBB,RRBR,RRRB,RRRR},equally likely

Accepted Answer

This problem can be modeled as a Markov Chain. Each state represents the number of consecutive red cards that have been drawn, and there are two absorbing states: one when a black card is drawn and one when four consecutive red cards are drawn. The transition probabilities depend on the current state.

The state diagram for choice D is as follows:

START -> R -> RR -> RRR -> RRRR (absorbing state)
                       -> B (absorbing state)
              -> START (with probability 1/2)

Here, the probability of going from any state to a black card is 1/2, and the probability of going from any state to the next state with a red card is 1/2. Therefore, the transition probabilities do not depend on the current state.

Choice D is therefore the best choice.

Question 3

A random spinner can land on red,green,blue,or yellow.If on the first three spins it lands once each on red,green,and yellow,is it more likely to land on blue on the fourth spin?
A)Yes,because every color is equally likely to occur.
B)Yes,because the Probability Assignment Rule dictates that all outcomes should occur.
C)No,because the spins are disjoint events.
D)Yes,because the spinner shows randomness,not chaos.
E)No,because knowing one outcome will not affect the next.

Accepted Answer

The outcome of each spin is independent of the previous spins, meaning that the probability of landing on any given color remains the same for each spin, regardless of the outcomes of the previous spins.

Question 4

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).Aces count as 1 or 11.Kings count as 10.You are interested in the total count of the two cards,with a maximum count of 21 (that is,AsAh = 12).
A){12,19,20,21},equally likely
B){12,19,20,21,22},not equally likely
C){12,19,20,21,22},equally likely
D){12,19,20,21},not equally likely
E)19,20,21,22},not equally likely

Accepted Answer

The possible combinations of two cards and their total counts are: AsAh (12), AsKs or As9h (either 11+10 or 11+9, so 21), AhKs or Ah9h (also 21), and Ks9h (19). There are no combinations that would result in a total count of 22 because the maximum value for an Ace is considered 11, and the other cards do not combine to exceed 21. The combinations are not equally likely because there are more ways to achieve a total count of 21 (four ways: AsKs, As9h, AhKs, Ah9h) compared to the other totals (one way each for 12 and 19).

Question 5

Toss a fair coin three times; record the order of heads and tails.
A){HHH,HHT,TTH,TTT},equally likely
B){HHH,HHT,HTH,HHT,HTT,THT,TTH,TTT},equally likely
C){HHH,HHT,TTH,TTT},not equally likely
D){H,T},equally likely
E){HHH,HHT,HTH,HHT,HTT,THT,TTH,TTT},not equally likely

Accepted Answer

Each toss of a fair coin has two outcomes (H or T), so with three tosses, there are 2^3 = 8 possible outcomes, all equally likely.

Question 6

What is the probability of an event?
A)A number between 0 and 1 that reports the likelihood of the event's occurrence
B)A collection of outcomes
C)A single attempt or realization of a random phenomenon
D)Its long-run relative frequency
E)Two of the above

Accepted Answer

A and D are correct because the probability of an event is a measure between 0 and 1 that indicates how likely an event is to occur, and it can also be described as the event's long-run relative frequency after many trials.

Question 7

A telemarketer is almost through with her entire shift,and she has not made a single sale.Her manager says that she has a good chance of making a sale on her last few phone calls because she is due.Is her manager right?
A)No,because the Probability Assignment Rule applies in the long-run,not in the short-run.
B)Yes,because the Law of Averages is valid for independent events.
C)No,because there is no Law of Large Numbers for independent events.
D)Yes,because the Law of Large Numbers is valid for independent events.
E)No,because there is no Law of Averages for independent events.

Accepted Answer

The Law of Averages is a misconception; independent events do not influence each other, so past performance does not affect future outcomes.

Question 8

Toss a fair coin five times; record the number of heads.
A){0,1,2,3,4,5},equally likely
B){0,1,2,3,4,5},not equally likely
C){1,2,3,4,5},equally likely
D){1,2,3,4,5},not equally likely
E){1,2},equally likely

Accepted Answer

When tossing a fair coin five times, all outcomes from 0 heads (all tails) to 5 heads (all heads) are possible, making the set {0,1,2,3,4,5} correct. However, these outcomes are not equally likely due to the different combinations that can result in each number of heads. For example, there is only 1 way to get 0 heads (all tails) and 1 way to get 5 heads (all heads), but there are more ways to get 2 or 3 heads, making some outcomes more likely than others.

Question 9

Roll a fair 6-sided die; record the number.
A){1,2,3,4,5,6},not equally likely
B){1,6},equally likely
C){1,2,3,4},equally likely
D){1,6},not equally likely
E){1,2,3,4,5,6},equally likely

Accepted Answer

Rolling a fair 6-sided die has equally likely outcomes of 1, 2, 3, 4, 5, and 6, making choice E the best option.

Question 10

Roll two fair 6-sided dice; record the smaller number.
A){1,2,3,4,5,6},not equally likely
B){1,2,3,4,5},equally likely
C){0,1,2,3,4,5},equally likely
D){1,2,3,4,5,6},equally likely
E){1,2,3,4,5},not equally likely

Accepted Answer

When rolling two fair 6-sided dice and recording the smaller number, the possible outcomes are indeed {1,2,3,4,5,6}, but they are not equally likely. For example, there is only one way to get a smaller number of 1 when both dice show a 1, but there are more ways to get a smaller number of 2 (one die shows a 2 and the other shows a 2 or higher).

Question 11

At the horse racing track,a gambler bets on the wrong horse in a 10-horse field nine times in a row.Later,when talking to a friend,he said he was confident that he would pick the winner the next time,because he was "due to pick a winner." Comment on his reasoning.
A)This is false reasoning because he doesn't appear to be lucky.
B)This is false reasoning because there is no Law of Averages for independent events.
C)When there are 10 horses in a race and he has chosen the wrong horse nine times in a row,he statistically should pick a winner the next time.
D)If he doesn't pick the winning horse the next time,he will shortly after that.
E)None of the above apply.

Accepted Answer

The gambler's reasoning is based on the false assumption that past events can influence future outcomes in independent events like picking the winning horse in each race. The Law of Averages does not apply in this case, and each race is a separate event with its own outcome, regardless of previous results. Therefore, the gambler's confidence in picking the winner of the next race is unfounded.

Question 12

Roll two fair 6-sided dice; record the positive difference.
A){0,6},not equally likely
B){0,1,2,3,4,5},not equally likely
C){1,2,3,4,5,6},equally likely
D){0,1,2,3,4,5},equally likely
E){1,2,3,4,5,6},not equally likely

Accepted Answer

The possible positive differences when rolling two fair 6-sided dice range from 0 (when both dice show the same number) to 5 (when one die shows 6 and the other shows 1, or vice versa). These differences are not equally likely because there are more ways to roll a smaller difference (e.g., there are 6 ways to roll a difference of 0, but only 1 way to roll a difference of 5).

Question 13

Roll a fair 6-sided die and record the number.Then flip a fair coin and record whether it comes up heads or tails.
A){1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T},equally likely
B){1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T},not equally likely
C){1,2,3,4,5,6,H,T},not equally likely
D){1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T,HH,HT,TH,TT},equally likely
E){1H,2T,3H,4T,5H,6T},equally likely

Accepted Answer

This is the only option where all outcomes are equally likely, as both the die and the coin are fair.

Question 14

A family has two children; record the genders in order of birth.
A){BB,BG,GG},equally likely
B){BB,BG,GB,GG},equally likely
C){BB,BG,GG},not equally likely
D){BB,BG,GB,GG},not equally likely
E){B,G},equally likely

Accepted Answer

This is the classic setup for a family with two children, in which each child can be either a boy (B) or a girl (G). There are four possible outcomes: BB (both boys), BG (one boy and one girl), GB (one girl and one boy), and GG (both girls). Each of these outcomes is equally likely.

Choice A is incorrect because it only includes three possible outcomes, when there are actually four.

Choice C is incorrect because it assumes that having a boy or a girl first is equally likely, when in fact the probability of having a boy or a girl first is always 1/2.

Choice D is incorrect because it includes GB and BG as separate outcomes, even though they are the same result (one girl and one boy).

Choice E is incorrect because it only lists the genders without considering the order of birth.

Question 15

The weather reporter predicts that there is a 20% chance of snow tomorrow for a certain region.What is meant by this phrase?
A)Snow occurs 20% of the time in this region.
B)The occurrence of snow is "truly random" and will occur 20% of the time.
C)In circumstances "like this," snow occurs 20% of the time.
D)20% of the time it snows on this date.
E)It will snow 20% of the day tomorrow.

Accepted Answer

The weather reporter is saying that in "circumstances like this," meaning the current weather patterns and conditions, snow occurs 20% of the time. This does not mean that it will definitely snow tomorrow, but rather that there is a small chance based on the current conditions. Option A and D are incorrect because they suggest a fixed probability, while B and E are incorrect because they do not accurately capture the probabilistic nature of the statement.

Question 16

The following contingency table provides a joint frequency distribution for a group of retired people by career and age at retirement. Age at Retirement $$\begin{array} { | c | c | c | c | c | c | } 
\hline \text { Career } & 50 - 55 & 56 - 60 & 61 - 65 & \text { Over 65 } & \text { Total } \
\hline \text { Attorney } & 12 & 44 & 73 & 43 & 172 \
\hline \begin{array} { c } 
\text { College } \
\text { Professor }
\end{array} & 9 & 45 & 71 & 33 & 158 \
\hline \text { Secretary } & 21 & 45 & 63 & 49 & 178 \
\hline \text { Store Clerk } & 18 & 44 & 70 & 50 & 182 \
\hline \text { Total } & 60 & 178 & 277 & 175 & 690 \
\hline
\end{array}$$ Suppose one of these people is selected at random.Compute the probability that the person selected was an attorney who retired between 61 and 65.
A)0.401
B)0.106
C)0.264
D)0.249
E)0.424

Accepted Answer

The probability that the person selected was an attorney who retired between 61 and 65 is calculated by dividing the number of attorneys who retired between 61 and 65 (73) by the total number of people (690). Thus, the probability is $ \frac{73}{690} \approx 0.106 $.

Question 17

Roll a fair 6-sided die eight times; record the length of the longest run of sixes.
A){0,1,2,3,4,5,6,7,8},not equally likely
B){0,1,2,3,4,5,6,7,8},equally likely
C){0,1,2,3,4,5,6},not equally likely
D){1,2,3,4,5,6,7,8},not equally likely
E){1,2,3,4,5,6},equally likely

Accepted Answer

The set of possible lengths for the longest run of sixes ranges from 0 (if no sixes are rolled) to 8 (if all rolls are sixes), making choice A correct. However, these outcomes are not equally likely due to the varying probabilities of rolling consecutive sixes.

Question 18

The following contingency table provides a joint frequency distribution for a group of retired people by career and age at retirement. Age at Retirement $$\begin{array} { | c | c | c | c | c | c | } 
\hline \text { Career } & 50 - 55 & 56 - 60 & 61 - 65 & \text { Over 65 } & \text { Total } \
\hline \text { Attorney } & 8 & 41 & 87 & 36 & 172 \
\hline \begin{array} { c } 
\text { College } \
\text { Professor }
\end{array} & 10 & 38 & 71 & 31 & 150 \
\hline \text { Secretary } & 21 & 45 & 63 & 49 & 178 \
\hline \text { Store Clerk } & 18 & 44 & 70 & 50 & 182 \
\hline \text { Total } & 57 & 168 & 291 & 166 & 682 \
\hline
\end{array}$$ Suppose one of these people is selected at random.Compute the probability that the person selected was a store clerk.
A)0.084
B)0.099
C)0.316
D)0.267
E)0.026

Accepted Answer

The probability that the person selected was a store clerk is calculated by dividing the total number of store clerks by the total number of people. There are 182 store clerks out of a total of 682 people, so the probability is $182 / 682 = 0.267$.

Question 19

When a weather forecaster predicts a 20% chance of thunderstorms,is this an empirical probability,theoretical probability,or personal probability?
A)Theoretical probability
B)Empirical probability
C)Personal probability
D)None of the above
E)All of the above

Accepted Answer

Empirical probability is based on observed data, and weather forecasts use historical weather data to predict future events, making it an empirical probability.

Question 20

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).
A){AsAh,AsKs,As9h,AhKs,Ah9h,Ks9h},not equally likely
B){AsAh,AsKs,As9h,AhKs,Ah9h,Ks9h},equally likely
C){AsAs,AsAh,AsKs,As9h,AhAh,AhKs,Ah9h,KsKs,Ks9h,9h9h},not equally likely
D){As,Ah,Ks,9h},equally likely
E){AsAs,AsAh,AsKs,As9h,AhAh,AhKs,Ah9h,KsKs,Ks9h,9h9h},equally likely

Accepted Answer

The answer of Consider a game that consists of dealing...

Quiz 12: From Randomness to Probability

Pick 5 cards from a randomly shuffled standard deck without replacement; record the number of eights.

Pick cards from a randomly shuffled deck until you get a black card or 4 consecutive red cards; record the cards in order.

A random spinner can land on red,green,blue,or yellow.If on the first three spins it lands once each on red,green,and yellow,is it more likely to land on blue on the fourth spin?

Toss a fair coin three times; record the order of heads and tails.

What is the probability of an event?

A telemarketer is almost through with her entire shift,and she has not made a single sale.Her manager says that she has a good chance of making a sale on her last few phone calls because she is due.Is her manager right?

Toss a fair coin five times; record the number of heads.

Roll a fair 6-sided die; record the number.

Roll two fair 6-sided dice; record the smaller number.

At the horse racing track,a gambler bets on the wrong horse in a 10-horse field nine times in a row.Later,when talking to a friend,he said he was confident that he would pick the winner the next time,because he was "due to pick a winner." Comment on his reasoning.

Roll two fair 6-sided dice; record the positive difference.

Roll a fair 6-sided die and record the number.Then flip a fair coin and record whether it comes up heads or tails.

A family has two children; record the genders in order of birth.

The weather reporter predicts that there is a 20% chance of snow tomorrow for a certain region.What is meant by this phrase?

Roll a fair 6-sided die eight times; record the length of the longest run of sixes.

When a weather forecaster predicts a 20% chance of thunderstorms,is this an empirical probability,theoretical probability,or personal probability?

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As) ,the Ace of Hearts (Ah) ,the King of Spades (Ks) and the 9 of Hearts (9h) .