Question 1

Provide an appropriate response.
-An insurance company sells an insurance policy for $1000. If there is no claim on a policy, the company makes a profit of $1000. If there is a claim on a policy, the company faces a large loss on
That policy. The expected value to the company, per policy, is $250.
Which of the following statements is (are)true? A: The most likely outcome on any single policy is a profit for the company of $\$ 250$.
B: If the company sells only a few policies, its profit is hard to predict.
C: If the company sells a large number of policies, the average profit per policy will be close to \$250.

Accepted Answer

B: The nature of probability and large numbers means that with only a few policies, the variance in outcomes can be significant, making profits unpredictable. C: The law of large numbers suggests that as the number of policies sold increases, the average profit per policy will approach the expected value of $250.

Question 2

Provide an appropriate response.
-Which of the following statements makes sense? A: When a balanced die is rolled, there are 6 possible outcomes, therefore the probability that I roll a six is 1/6

B: On my test there are four possible outcomes, I could get an $\mathrm { A }$, a B, a C, or I could fail. Therefore the probability that I get an $\mathrm { A }$ is $1 / 4$

C: When I flip two coins there are three possible outcomes: 0 tails, 1 tail, or 2 tails. Therefore the probability that I will get 1 tail is $1 / 3$

Accepted Answer

A) The statement about rolling a die is correct because a fair six-sided die has six equally likely outcomes, making the probability of rolling any specific number (such as a six) 1/6. B) The statement about the test outcomes incorrectly assumes equal probability for each grade without considering performance or grading criteria. C) The statement about flipping two coins simplifies the outcomes into categories based on the number of tails, which is a valid approach, and calculates the probability based on these simplified outcomes.

Question 3

Provide an appropriate response.
-Chantal noted that there are two equally likely outcomes when you flip a fair coin and heads is one of those outcomes. She concluded that the probability of getting heads on a flip of a fair coin is 1/2 .
Which method did Chantal use?

Accepted Answer

Chantal used the theoretical method, which involves calculating probabilities based on the known possible outcomes without conducting an experiment or observing the outcomes. Since a fair coin has two sides and both are equally likely, the theoretical probability of getting heads is 1/2.

Question 4

Provide an appropriate response.
-A card is selected at random from a standard deck of 52 cards. Let $\mathrm { A } =$ event the card is a heart
$B =$ event the card is a red card Which of the following is (are)true? $A : P ( A$ or $B ) = P ( A )$
B: $\mathrm { P } ( \mathrm { A }$ or $\mathrm { B } ) = \mathrm { P } ( \mathrm { B } )$
$C : P ( A$ and $B ) = P ( B )$
D: $\mathrm { P } ( \mathrm { A }$ and $\mathrm { B } ) = \mathrm { P } ( \mathrm { A } )$

Accepted Answer

B: Since all hearts are red cards, $P(A \text{ or } B) = P(B)$ because $A$ is a subset of $B$. D: $P(A \text{ and } B) = P(A)$ because every heart is a red card, so the probability of selecting a heart (A) is the same as the probability of selecting a card that is both a heart and a red card.

Question 5

Provide an appropriate response.
-The permutation formula can be used to determine which of the following? A: The number of ways of completing a test consisting of 10 true/false questions
B: The number of ways of arranging 8 people in a line
$C$ : The number of ways a panel of people could vote if each person votes yes or no

Accepted Answer

The permutation formula is used to determine the number of ways of arranging a set of objects in a specific order, which applies to arranging 8 people in a line (B). Choices A and C involve scenarios where order matters less and are more suited to combinations or other counting principles.

Question 6

Provide an appropriate response.
-When using the counting rules to count the number of possible outcomes, the combinations formula can be used in which of the following situations?

Accepted Answer

The combinations formula is used when repetitions are not allowed and the order of arrangement is unimportant, which allows for the calculation of how many ways a given number of items can be selected from a larger pool when the order doesn't matter.

Question 7

Provide an appropriate response.
-Which of the following are examples of independent events?

Accepted Answer

The answer of Provide an appropriate response.
-Which of the following...

Question 8

Provide an appropriate response.
-Event A is that it rains in Santa Cruz tomorrow. Event B is that rains in Paris, France tomorrow. Are events A and B: A: Overlapping, Independent?
B: Overlapping, Dependent?
C: Non-overlapping, Independent?
D: Non-overlapping, Dependent?

Accepted Answer

Events A and B are non-overlapping because they occur in different geographical locations, but they are independent since the occurrence of rain in one location does not affect the likelihood of rain in the other location.

Question 9

Provide an appropriate response.
-A game involves tossing a fair coin 200 times. For each tail that appears you lose $1. For each head that appears, you win $1.
After 100 tosses, there have been 40 heads and 60 tails. At this point, with 100 tosses to go, what
Can you conclude?
A: You are more likely to lose than to win.
B: You are still equally likely to win or lose because in the long run there should be 50% heads.
C: In the next 100 tosses you are likely to get more heads than tails.

Accepted Answer

The answer of Provide an appropriate response.
-A game involves tossing...

Question 10

Provide an appropriate response.
-The combination formula can be used to determine which of the following? A: The number of sequences of 3 letters using the letters $A , B , C$, and D if repetitions are not allowed
B: The number of ways of choosing a president and a treasurer from 10 people C: The number of ways to choose 3 friends from 10 to invite to dinner

Accepted Answer

The answer of Provide an appropriate response.
-The combination formula can...

Question 11

Provide an appropriate response.
-Suppose that S and T are mutually exclusive events. Which of the following statements is true?

Accepted Answer

Mutually exclusive events cannot occur at the same time, meaning if one event happens, the other cannot. This directly contradicts the definition of independent events, where the occurrence of one event does not affect the probability of the other occurring. Therefore, mutually exclusive events cannot be independent.

Question 12

Provide an appropriate response.
-Suppose that 4 items are to be selected from N items and that repetitions are not allowed. Which of the following is true?

Accepted Answer

Permutations consider the order of selection, whereas combinations do not. When selecting 4 items from N without repetition, the number of permutations is N!/(N-4)!, and the number of combinations is N!/[4!(N-4)!]. The ratio of permutations to combinations is therefore 4! = 24, meaning there are 24 times as many permutations as combinations.

Question 13

Provide an appropriate response.
-Event A is that Lisa votes for Candidate A in the gubernatorial election and event B is that she votes for candidate B. Are events A and B: A: Overlapping, Independent?
B: Overlapping, Dependent?
$\mathrm { C } :$ Non-overlapping, Independent?
D: Non-overlapping, Dependent?

Accepted Answer

The answer of Provide an appropriate response.
-Event A is that...

Question 14

Provide an appropriate response.
-When using the counting rules to count the number of possible outcomes, the permutation formula can be used in which of the following situations?

Accepted Answer

The permutation formula is used when repetitions are not allowed and the order of arrangement is important, as it calculates the number of ways to arrange a subset of items from a larger set where the order matters.

Question 15

Provide an appropriate response.
-When a balanced die is rolled, the probability that a four will appear is $\frac { 1 } { 6 }$ Which of the following statements is a reasonable conclusion? A: If I roll a balanced die 6 times I will get one four
B: If I roll a balanced die 300 times I will get fifty fours
C: If I roll a balanced die 1200 times, I will get approximately 200 fours

Accepted Answer

The probability of rolling a four is $\frac{1}{6}$. Over a large number of rolls, the expected number of fours is the total rolls multiplied by $\frac{1}{6}$. Thus, rolling 1200 times gives an expected 200 fours, making C reasonable. A and B are not reasonable because they imply certainty rather than probability.

Question 16

Provide an appropriate response.
-Which of the following events has a probability of 1? A: I will get a perfect score on my math test if I study
B: I will die some day
C: The world will go on turning if I fail my test

Accepted Answer

The probability of dying someday is 1, as it is a certainty for all living beings. The world continuing to turn regardless of a test outcome is also certain, making its probability 1.

Question 17

Provide an appropriate response.
-When a balanced die is rolled, the probability that a six will appear is $\frac { 1 } { 6 } .$ . Suppose that you have just rolled the die five times without getting a six. What is the probability that you will get a six on
The next roll of the die?

Accepted Answer

The probability of rolling a six on a fair die is always $\frac{1}{6}$, regardless of previous rolls, due to the independence of each roll.

Question 18

Provide an appropriate response.
-When tossing a fair coin, which of the following events is more unlikely? A: Getting $60 \%$ tails in 10 tosses
B: Getting $60 \%$ tails in 100 tosses
$\mathrm { C }$ : Getting $60 \%$ tails in 1000 tosses

Accepted Answer

As the number of tosses increases, the relative frequency of getting tails tends to stabilize closer to the expected probability of $50\%$. Therefore, getting $60\%$ tails becomes more unlikely as the number of tosses increases, making option C (1000 tosses) the most unlikely scenario.

Question 19

Provide an appropriate response.
-In a large casino, the house wins on its blackjack games with a probability of 50.7%. Which of the following events is the most unlikely?
A: You win at a single game.
B: You come out ahead after playing forty times.
C: You win at a single game given that you have just won ten games in a row.
D: You win at a single game given that you have just lost ten games in a row.

Accepted Answer

The most unlikely event is coming out ahead after playing forty times due to the house's edge of 50.7%, which means the probability of winning decreases with the number of games played. Winning or losing previous games does not affect the outcome of a new game due to the independence of each game.

Question 20

Provide an appropriate response.
-Melissa estimates that the probability that she will marry her boyfriend is 0.6. Which method for determining probabilities did she use?

Accepted Answer

Melissa used the subjective method, as this method involves personal judgment or estimation rather than objective calculation or physical measurement.

Quiz 7: Probability: Living With the Odds

Provide an appropriate response. -Chantal noted that there are two equally likely outcomes when you flip a fair coin and heads is one of those outcomes. She concluded that the probability of getting heads on a flip of a fair coin is 1/2 . Which method did Chantal use?

Provide an appropriate response. -When using the counting rules to count the number of possible outcomes, the combinations formula can be used in which of the following situations?

Provide an appropriate response. -Which of the following are examples of independent events?

Provide an appropriate response. -Event A is that it rains in Santa Cruz tomorrow. Event B is that rains in Paris, France tomorrow. Are events A and B: A: Overlapping, Independent? B: Overlapping, Dependent? C: Non-overlapping, Independent? D: Non-overlapping, Dependent?

Provide an appropriate response. -Suppose that S and T are mutually exclusive events. Which of the following statements is true?

Provide an appropriate response. -Suppose that 4 items are to be selected from N items and that repetitions are not allowed. Which of the following is true?

Provide an appropriate response. -When using the counting rules to count the number of possible outcomes, the permutation formula can be used in which of the following situations?

Provide an appropriate response. -Which of the following events has a probability of 1? A: I will get a perfect score on my math test if I study B: I will die some day C: The world will go on turning if I fail my test

Provide an appropriate response. -When a balanced die is rolled, the probability that a six will appear is 16.\frac { 1 } { 6 } .61​. . Suppose that you have just rolled the die five times without getting a six. What is the probability that you will get a six on The next roll of the die?

Provide an appropriate response. -When tossing a fair coin, which of the following events is more unlikely? A: Getting 60%60 \%60% tails in 10 tosses B: Getting 60%60 \%60% tails in 100 tosses C\mathrm { C }C : Getting 60%60 \%60% tails in 1000 tosses

Provide an appropriate response. -Melissa estimates that the probability that she will marry her boyfriend is 0.6. Which method for determining probabilities did she use?

Provide an appropriate response. -When a balanced die is rolled, the probability that a six will appear is $\frac { 1 } { 6 } .$ . Suppose that you have just rolled the die five times without getting a six. What is the probability that you will get a six on The next roll of the die?

Provide an appropriate response. -When tossing a fair coin, which of the following events is more unlikely? A: Getting $60 \%$ tails in 10 tosses B: Getting $60 \%$ tails in 100 tosses $\mathrm { C }$ : Getting $60 \%$ tails in 1000 tosses