Question 1

Describe an event whose probability of occurring is 1 and explain what that probability means. Describe an event whose probability of occurring is 0 and explain what that probability means.

Accepted Answer

The answer of Describe an event whose probability of occurring...

Question 2

Find the indicated probability by using the complementation rule.
-The distribution of B.A. degrees conferred by a local college is listed below, by major. 
$\begin{array} { l r } 
\underline { \text { Major } } & \text { Frequency } \
\text { English } & 2073 \
\text { Mathematics } & 2164 \
\text { Chemistry } & 318 \
\text { Physics } & 856 \
\text { Liberal Arts } & 1358 \
\text { Business } & 1676 \
\text { Engineering } & \frac { 868 } { 9313 }
\end{array}$ 
What is the probability that a randomly selected degree is not in Mathematics? 
A)0.303
B)0.682
C)0.768
D)0.232

Accepted Answer

The probability of selecting a Mathematics degree is 2164/9313. Using the complementation rule, the probability of not selecting a Mathematics degree is 1 - (2164/9313) = 0.768.

Question 3

Suppose a student is taking a 5-response multiple choice exam; that is, the choices are A, B, C, D, and E, with only one of the responses correct. Describe the complement method for determining the probability of getting at least one of the questions correct on the 15-question exam. Why would the complement method be the method of choice for this problem?

Accepted Answer

The answer of Suppose a student is taking a 5-response...

Question 4

Construct a Venn diagram representing the event $( ( A \& B ) \text { or } C )$

Accepted Answer

The answer of Construct a Venn diagram representing the event...

Question 5

Discuss the differences, both in applications and in the formulas, for combinations and permutations. Give an example of each.

Accepted Answer

The answer of Discuss the differences, both in applications and...

Question 6

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, $\mathrm { P } \left( \mathrm { P } _ { 2 } \mid \mathrm { R } _ { 1 } \right) = 0.553$. Interpret this probability in terms of percentages.

Accepted Answer

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

Question 7

Solve the problem.
-A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four cards of one suit and one card of another suit? 
A)37,180
B)9295
C)111,540
D)715

Accepted Answer

The number of ways to choose a suit for the four cards is 4 (since there are 4 suits). For each of these choices, there are $ \binom{13}{4} $ ways to choose the four cards from the chosen suit. Then, there are 48 remaining cards (from the other three suits), and $ \binom{48}{1} $ ways to choose one card from these. So, the total number of different hands is $ 4 \times \binom{13}{4} \times \binom{48}{1} = 4 \times 715 \times 48 = 111,540 $.

Question 8

Use the rule of total probability to find the indicated probability.
-A company is conducting a sweepstakes, and ships two boxes of game pieces to a particular store. Box A has 4% of its contents being winners, while 6% of the contents of box B are winners. Box A
Contains 34% of the total tickets. If the contents of both boxes are mixed in a drawer and a ticket is
Chosen at random, what is the probability it is a winner? 
A)0.1
B)0.053
C)0.05
D)0.014

Accepted Answer

The probability of picking a winner is calculated using the rule of total probability, which in this case is the sum of the probabilities of picking a winner from each box, weighted by the probability of picking from that box. For Box A, with 34% of the tickets and a 4% win rate, the contribution is 0.34 * 0.04. For Box B, with 66% of the tickets (100% - 34%) and a 6% win rate, the contribution is 0.66 * 0.06. Adding these together gives 0.0136 + 0.0396 = 0.0532, which rounds to 0.053.

Question 9

The age distribution of students at a community college is given below. $\begin{array}{lc}
\text { Age (years) } & \text { Number of students (f) } \
\hline \text { Under 21 } & 412 \
21-24 & 416 \
25-28 & 274 \
29-32 & 157 \
33-36 & 103 \
37-40 & 54 \
\text { Over 40 } & 95 \
\hline & 1511
\end{array}$
 A student from the community college is selected at random. Find the probability that the student is under 37 years old. Give your answer as a decimal rounded to three decimal places. 
A)0.099
B)0.901
C)0.036
D)0.068

Accepted Answer

To find the probability that a randomly selected student is under 37 years old, add the number of students in all age groups under 37 years old and divide by the total number of students. This includes the groups under 21, 21-24, 25-28, 29-32, and 33-36, which sum to 412 + 416 + 274 + 157 + 103 = 1362. The total number of students is 1511. Thus, the probability is 1362 / 1511 = 0.901 (rounded to three decimal places).

Question 10

Determine whether the events are independent.
-The following contingency table provides a joint frequency distribution for a random sample of patients at a hospital classified by blood type and sex. 
Suppose one of the patients is selected at random. Are the events $\mathrm { T } _ { 2 }$ and $\mathrm { S } _ { 2 }$ independent?

Accepted Answer

The answer of Determine whether the events are independent.
-The following...

Question 11

$\text { Consider the following formulas: } ( n ) _ { r } = \frac { n ! } { ( n - r ) ! } \text { and } \left( \begin{array} { l }  n \ r \end{array} \right) = \frac { n ! } { ( n - r ) ! r ! } \text {. }$
 Given the same values for n and r in each formula, which is the smaller value, P or C? How does this relate to the concept of counting the number of outcomes based on whether or not order is a criterion?

Accepted Answer

The answer of \(\text { Consider the following formulas: }...

Question 12

Describe the specified event in words.
-The number of hours needed by sixth grade students to complete a research project was recorded with the following results. $$\begin{array} { c c } 
\text { Hours } & \text { Number of students (f) } \
\hline 4 & 15 \
5 & 11 \
6 & 19 \
7 & 6 \
8 & 9 \
9 & 16 \
10 & 2
\end{array}$$ A student is selected at random. The event A is defined as follows.
A = the event the student took between 5 and
9 hours inclusive
B = the event the student took at least 6 hours
Describe the event (A & B) in words.
A)The event the student took between 6 and 9 hours inclusive
B)The event the student took between 5 and 6 hours inclusive
C)The event the student at least 5 hours
D)The event the student took more than 6 hours and less than 9 hours

Accepted Answer

Event A is described as the student taking between 5 and 9 hours inclusive, and Event B is described as the student taking at least 6 hours. The overlap of these two events (A & B) is the event where the student took between 6 and 9 hours inclusive.

Question 13

Find the indicated probability.
-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.

A person who voted in the presidential election is selected at random. Compute the probability that the person selected voted Democrat.  
A)$0.442$
B) $0.406$
C) $0.241$
D) $0.098$

Accepted Answer

The answer of Find the indicated probability.
-The following contingency table...

Question 14

Find the indicated probability by using the special addition rule.
-The distribution of B.A. degrees conferred by a local college is listed below, by major. 
$\begin{array} { l r } 
\underline { \text { Major } } & \text { Frequency } \
\text { English } & 2073 \
\text { Mathematics } & 2164 \
\text { Chemistry } & 318 \
\text { Physics } & 856 \
\text { Liberal Arts } & 1358 \
\text { Business } & 1676 \
\text { Engineering } & \underline { 868 } \
& 9313
\end{array}$ 
What is the probability that a randomly selected degree is in English or Mathematics? 
A)0.455
B)0.424
C)0.010
D)0.517

Accepted Answer

The probability of selecting an English or Mathematics degree is the sum of the probabilities of selecting each individually, which is $\frac{2073}{9313} + \frac{2164}{9313} = 0.455$.

Question 15

Suppose that you roll a die and record the number that comes up and then flip a coin and record whether it comes up heads or tails. One possible outcome can be represented as 2H (a two on the die followed by heads). Make a list of all the possible outcomes. What is the probability that you get tails and an even number? What assumption are you making when you find this probability?

Accepted Answer

The answer of Suppose that you roll a die and...

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 a pair of events B and C for this experiment such that the events A and
B are mutually exclusive but the collection of events A, B, and C is not mutually exclusive.

Accepted Answer

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

Question 17

Determine whether the events are independent.
-In a local election, 53.4% of those aged under 40 and 47.3% of those aged over 40 vote in favor of a certain ballot measure. Are age and position on the ballot measure independent?

Accepted Answer

The percentage of people voting in favor of the ballot measure differs between the two age groups, indicating that the likelihood of supporting the measure is affected by age, thus showing that age and position on the ballot measure are not independent.

Question 18

A relative frequency distribution is given below for the size of families in one U.S. city. 
$\begin{array} { l c } 
\text { Size } & \text { Relative frequency } \
\hline 2 & 0.458 \
3 & 0.205 \
4 & 0.199 \
5 & 0.086 \
6 & 0.034 \
7 + & 0.018
\end{array}$
 A family is selected at random. Find the probability that the size of the family is at most 6. Round approximations to three decimal places. 
A)0.034
B)0.948
C)0.052
D)0.982

Accepted Answer

The answer of A relative frequency distribution is given below...

Question 19

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

Accepted Answer

The answer of Construct a Venn diagram portraying three events...

Question 20

Find the indicated probability by using the special addition rule.
-A relative frequency distribution is given below for the size of families in one U.S. city. 
$\begin{array} { l c } 
\text { Size } & \text { Relative frequency } \
\hline 2 & 0.416 \
3 & 0.217 \
4 & 0.210 \
5 & 0.102 \
6 & 0.040 \
7 + & 0.015
\end{array}$ 
A family is selected at random. Find the probability that the size of the family is between 2 and 5 inclusive. Round approximations to three decimal places. 
A)0.843
B)0.945
C)0.518
D)0.427

Accepted Answer

The answer of Find the indicated probability by using the...

Quiz 4: Probability Concepts

Describe an event whose probability of occurring is 1 and explain what that probability means. Describe an event whose probability of occurring is 0 and explain what that probability means.

Construct a Venn diagram representing the event ((A&B) or C)( ( A \& B ) \text { or } C )((A&B) or C)

Discuss the differences, both in applications and in the formulas, for combinations and permutations. Give an example of each.

Solve the problem. -A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four cards of one suit and one card of another suit?

Determine whether the events are independent. -In a local election, 53.4% of those aged under 40 and 47.3% of those aged over 40 vote in favor of a certain ballot measure. Are age and position on the ballot measure independent?

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

Construct a Venn diagram representing the event $( ( A \& B ) \text { or } C )$