Question 1

If $\mathrm { P } ( \mathrm { A } ) = 0.25 , \mathrm { P } ( \mathrm { B } ) = 0.8$, and $\mathrm { P } ( \mathrm { A } \& \mathrm {~B} ) = 0.23$, are $\mathrm { A }$ and $\mathrm { B }$ independent events? How can you tell?

Accepted Answer

$$\text { No, since } \mathrm { P } ( \mathrm { A } ) \cdot \mathrm { P } ( \mathrm { B } ) \neq \mathrm { P } ( \mathrm { A } \& \mathrm {~B} )$$

Question 2

Consider the following counting problem. Eight women and seven men are waiting in line at a movie theater. How many ways are there to arrange these 15 people amongst themselves such that
The eight women occupy the first eight places and the seven men the last seven places?
To solve this problem, which of the following rules would you use? 
A)Both the combinations rule and the basic counting rule
B)The basic counting rule only
C)The permutations rule only
D)Both the permutations rule and the basic counting rule

Accepted Answer

The permutations rule is used because the order of the people matters (arranging them in a line), and the basic counting rule is applied to multiply the number of ways to arrange the women and the men separately.

Question 3

Construct a Venn diagram portraying four events A, B, C, and D such that the collection of events A, B, and C is mutually exclusive, the collection of events A, B, and D is mutually exclusive, but the collection of events A, B, C, and D is not mutually exclusive.

Accepted Answer

To construct a Venn diagram that satisfies the given conditions, we need to understand the requirements:

1. The collection of events A, B, and C is mutually exclusive, which means that these events cannot occur at the same time. In other words, there is no overlap between A, B, and C in the Venn diagram.
2. The collection of events A, B, and D is mutually exclusive, which similarly means that A, B, and D cannot occur at the same time, and there is no overlap between them in the Venn diagram.
3. The collection of events A, B, C, and D is not mutually exclusive, which means that there is some overlap among these four events.

Given these conditions, we can infer that the overlap must involve event D with either C or both. Since A, B, and C are mutually exclusive, and A, B, and D are mutually exclusive, D cannot overlap with A or B. Therefore, D must overlap with C.

Here's how to draw the Venn diagram:

1. Draw three non-overlapping circles and label them A, B, and C. This represents the mutual exclusivity of A, B, and C.
2. Draw a fourth circle for event D. This circle should not overlap with A or B, but it should overlap with C to represent that D can occur with C, but not with A or B.

The resulting Venn diagram will show four circles where A, B, and C are separate, and D overlaps only with C. This satisfies the condition that A, B, and C as well as A, B, and D are mutually exclusive, but A, B, C, and D are not mutually exclusive because C and D can occur together.

Here's a textual representation of the Venn diagram:

```
   A       B
   ( )    ( )
      C       D
     ( )    ( )
```

In this diagram, the circles for A and B are separate, the circle for C is separate from A and B, and the circle for D overlaps with C but not with A or B.

Question 4

Explain why an event and its complement are always mutually exclusive and exhaustive.

Accepted Answer

The answer of Explain why an event and its complement...

Question 5

Suppose that S and T are mutually exclusive events. Which of the following statements is true? 
A)S and T must also be independent.
B)S and T cannot possibly be independent.
C)S and T may or may not be independent.

Accepted Answer

Mutually exclusive events cannot occur at the same time, and if one event occurs, it completely rules out the occurrence of the other. This dependency between the events means they cannot be independent. Independence implies that the occurrence of one event does not affect the probability of the other event occurring, which contradicts the definition of mutually exclusive events.

Question 6

Discuss the range of possible values for probabilities. Give examples to support each.

Accepted Answer

The answer of Discuss the range of possible values for...

Question 7

Interpret the symbol $\mathrm { P } ( \mathrm { B } \mid \mathrm { A } )$ and explain what is meant by the expression. What do we know if $P ( B \mid A )$ is not the same as $P ( B )$ ?

Accepted Answer

The answer of Interpret the symbol \(\mathrm { P }...

Question 8

Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books?
To solve this problem which of the following rules would you use? 
A)The combinations rule only
B)The basic counting rule only
C)Both the permutations rule and the basic counting rule
D)The permutations rule only

Accepted Answer

The correct rule to use is the combinations rule because the order in which Allison picks the books does not matter; what matters is which books are chosen.

Question 9

Consider the following counting problem. A pool of possible jurors consists of 11 men and 13 women. How many different juries consisting of 5 women and 7 men are possible?
To solve this problem, which of the following rules would you use? 
A)Both the combinations rule and the basic counting rule
B)The combinations rule only
C)Both the permutations rule and the basic counting rule
D)The basic counting rule only

Accepted Answer

To form a jury, you need to choose 5 women from 13 and 7 men from 11, which involves combinations. Then, you multiply the number of ways to choose women by the number of ways to choose men, which involves the basic counting rule.

Question 10

Give an example of a collection of events that are both mutually exclusive and exhaustive

Accepted Answer

The answer of Give an example of a collection of...

Question 11

The following contingency table provides a joint frequency distribution for the popular votes cast in the 1984 presidential election by region and political party. Data are in thousands, rounded to the nearest thousand.   If a person who voted in the 1984 presidential election is selected at random, $\mathrm { P } \left( \mathrm { R } _ { 2 } \& \mathrm { P } _ { 1 } \right) = 0.113$. Interpret this probability in terms of percentages.

Accepted Answer

The answer of The following contingency table provides a joint...

Question 12

Define mutually exclusive events and independent events. Give an example of each.

Accepted Answer

The answer of Define mutually exclusive events and independent events....

Question 13

A card is selected randomly from a standard deck of 52 cards. Let A = event that the card is an ace. Give examples of events B, C, and D such that A and B are independent, A and C are dependent but not mutually exclusive, and A and D are mutually exclusive.

Accepted Answer

The answer of A card is selected randomly from a...

Question 14

On an exam question asking for a probability, Sue had an answer of $\frac { 13 } { 8 }$ . Explain how she knew that this result was incorrect.

Accepted Answer

The answer of On an exam question asking for a...

Question 15

Consider the following counting problem. How many different sequences of 3 letters can be formed using the letters a, b, c, d, e if repetition is allowed?
To solve this problem which of the following rules would you use? 
A)The basic counting rule only
B)The combinations rule only
C)Both the permutations rule and the basic counting rule
D)The permutations rule only

Accepted Answer

Since repetition is allowed and the order of the letters matters, the basic counting rule is sufficient. For each of the 3 positions in the sequence, there are 5 choices (a, b, c, d, e), leading to 5^3 different sequences.

Question 16

An experiment consists of randomly selecting a card from a deck of 52. The event A is defined as follows.
A = event the card selected is a diamond
Give an example of an event B for this experiment such that the events A and B are mutually exclusive.

Accepted Answer

The answer of An experiment consists of randomly selecting a...

Question 17

Suppose that in an election for governor of Oregon there are five candidates of whom two are women. A statistics student reasons as follows. The probability that a woman will win the election is equal to $\frac { \mathrm { f } } { \mathrm { N } }$ which is $\frac { 2 } { 5 }$. What is wrong with his reasoning?

Accepted Answer

The answer of Suppose that in an election for governor...

Question 18

Interpret the following probability statement using the frequentist interpretation of probability. The probability is 0.83 that this particular type of surgery will be successful.

Accepted Answer

The answer of Interpret the following probability statement using the...

Question 19

What important question must you answer before computing an "or" probability? How does the answer influence your computation?

Accepted Answer

The answer of What important question must you answer before...

Question 20

The following contingency table provides a joint frequency distribution for the popular votes cast in the presidential election by region and political party. Data are in thousands, rounded to the nearest thousand.  If a person who voted in the presidential election is selected at random, $P \left( R _ { 2 } \mid P _ { 1 } \right) = 0.280$. Interpret this probability in terms of percentages.

Accepted Answer

The answer of The following contingency table provides a joint...

Quiz 4: Probability Concepts

If $\mathrm { P } ( \mathrm { A } ) = 0.25 , \mathrm { P } ( \mathrm { B } ) = 0.8$ , and $\mathrm { P } ( \mathrm { A } \& \mathrm {~B} ) = 0.23$ , are $\mathrm { A }$ and $\mathrm { B }$ independent events? How can you tell?

Construct a Venn diagram portraying four events A, B, C, and D such that the collection of events A, B, and C is mutually exclusive, the collection of events A, B, and D is mutually exclusive, but the collection of events A, B, C, and D is not mutually exclusive.

Explain why an event and its complement are always mutually exclusive and exhaustive.

Suppose that S and T are mutually exclusive events. Which of the following statements is true?

Discuss the range of possible values for probabilities. Give examples to support each.

Interpret the symbol $\mathrm { P } ( \mathrm { B } \mid \mathrm { A } )$ and explain what is meant by the expression. What do we know if $P ( B \mid A )$ is not the same as $P ( B )$ ?

Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books? To solve this problem which of the following rules would you use?

Consider the following counting problem. A pool of possible jurors consists of 11 men and 13 women. How many different juries consisting of 5 women and 7 men are possible? To solve this problem, which of the following rules would you use?

Give an example of a collection of events that are both mutually exclusive and exhaustive

Define mutually exclusive events and independent events. Give an example of each.

A card is selected randomly from a standard deck of 52 cards. Let A = event that the card is an ace. Give examples of events B, C, and D such that A and B are independent, A and C are dependent but not mutually exclusive, and A and D are mutually exclusive.

On an exam question asking for a probability, Sue had an answer of $\frac { 13 } { 8 }$ . Explain how she knew that this result was incorrect.

Consider the following counting problem. How many different sequences of 3 letters can be formed using the letters a, b, c, d, e if repetition is allowed? To solve this problem which of the following rules would you use?

An experiment consists of randomly selecting a card from a deck of 52. The event A is defined as follows. A = event the card selected is a diamond Give an example of an event B for this experiment such that the events A and B are mutually exclusive.

Interpret the following probability statement using the frequentist interpretation of probability. The probability is 0.83 that this particular type of surgery will be successful.

What important question must you answer before computing an "or" probability? How does the answer influence your computation?

Quiz 4: Probability Concepts

If P(A)=0.25,P(B)=0.8\mathrm { P } ( \mathrm { A } ) = 0.25 , \mathrm { P } ( \mathrm { B } ) = 0.8P(A)=0.25,P(B)=0.8 , and P(A& B)=0.23\mathrm { P } ( \mathrm { A } \& \mathrm {~B} ) = 0.23P(A& B)=0.23 , are A\mathrm { A }A and B\mathrm { B }B independent events? How can you tell?

Construct a Venn diagram portraying four events A, B, C, and D such that the collection of events A, B, and C is mutually exclusive, the collection of events A, B, and D is mutually exclusive, but the collection of events A, B, C, and D is not mutually exclusive.

Explain why an event and its complement are always mutually exclusive and exhaustive.

Suppose that S and T are mutually exclusive events. Which of the following statements is true?

Discuss the range of possible values for probabilities. Give examples to support each.

Interpret the symbol P(B∣A)\mathrm { P } ( \mathrm { B } \mid \mathrm { A } )P(B∣A) and explain what is meant by the expression. What do we know if P(B∣A)P ( B \mid A )P(B∣A) is not the same as P(B)P ( B )P(B) ?

Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books? To solve this problem which of the following rules would you use?

Consider the following counting problem. A pool of possible jurors consists of 11 men and 13 women. How many different juries consisting of 5 women and 7 men are possible? To solve this problem, which of the following rules would you use?

Give an example of a collection of events that are both mutually exclusive and exhaustive

Define mutually exclusive events and independent events. Give an example of each.

A card is selected randomly from a standard deck of 52 cards. Let A = event that the card is an ace. Give examples of events B, C, and D such that A and B are independent, A and C are dependent but not mutually exclusive, and A and D are mutually exclusive.

On an exam question asking for a probability, Sue had an answer of 138\frac { 13 } { 8 }813​ . Explain how she knew that this result was incorrect.

Consider the following counting problem. How many different sequences of 3 letters can be formed using the letters a, b, c, d, e if repetition is allowed? To solve this problem which of the following rules would you use?

An experiment consists of randomly selecting a card from a deck of 52. The event A is defined as follows. A = event the card selected is a diamond Give an example of an event B for this experiment such that the events A and B are mutually exclusive.

Interpret the following probability statement using the frequentist interpretation of probability. The probability is 0.83 that this particular type of surgery will be successful.

What important question must you answer before computing an "or" probability? How does the answer influence your computation?

If $\mathrm { P } ( \mathrm { A } ) = 0.25 , \mathrm { P } ( \mathrm { B } ) = 0.8$ , and $\mathrm { P } ( \mathrm { A } \& \mathrm {~B} ) = 0.23$ , are $\mathrm { A }$ and $\mathrm { B }$ independent events? How can you tell?

Interpret the symbol $\mathrm { P } ( \mathrm { B } \mid \mathrm { A } )$ and explain what is meant by the expression. What do we know if $P ( B \mid A )$ is not the same as $P ( B )$ ?

On an exam question asking for a probability, Sue had an answer of $\frac { 13 } { 8 }$ . Explain how she knew that this result was incorrect.