Question 1

Suzy grabs four marbles from a bag containing three white marbles and six blue marbles. Find the probability that two of the marbles are white, and two are blue.
 
A)  $\frac { 1 } { 6 }$
B)   $\frac { 1 } { 14 }$
C)   $\frac { 11 } { 42 }$
D)   $\frac { 5 } { 14 }$
E)  1

Accepted Answer

The probability of drawing two white marbles and two blue marbles can be calculated using combinations. The number of ways to choose two white marbles out of three is $C(3,2) = 3$, and the number of ways to choose two blue marbles out of six is $C(6,2) = 15$. The total number of ways to choose four marbles out of nine is $C(9,4) = 126$. Therefore, the probability is $\frac{3 \times 15}{126} = \frac{45}{126} = \frac{5}{14}$.

Question 2

A market survey shows that 50% of the population used Brand Z computers last year, 6% of the population quit their jobs last year, and 3% of the population used Brand Z computers and then quit their jobs. Are the events of using Brand Z computers and quitting your job independent 
 
Is a user of Brand Z computers more or less likely to quit a job than a randomly chosen person

Accepted Answer

The answer of A market survey shows that 50% of...

Question 3

A bag contains 4 red marbles, 2 green ones, 2 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she has at least 1 green marble.  
A) $\frac { 2 } { 9 }$
B) $\frac { 5 } { 6 }$
C) $\frac { 1 } { 5 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 9 }$

Accepted Answer

The probability of grabbing at least 1 green marble is easier to calculate by finding the complement, which is the probability of not grabbing any green marbles, and subtracting it from 1. There are 9 marbles that are not green out of a total of 9 marbles, so the probability of not picking a green marble in 5 tries is $\frac{7}{9} \times \frac{6}{8} \times \frac{5}{7} \times \frac{4}{6} \times \frac{3}{5} = \frac{1}{6}$. Therefore, the probability of picking at least one green marble is $1 - \frac{1}{6} = \frac{5}{6}$.

Question 4

In the New York State daily lottery game, a sequence of 6 (not necessarily different) digits in the range of 0-9 is selected at random. ​
Find the probability that all 6 are different. Round your answer to four decimal places.
​
A)0.8488
B) 0.0600
C) 0.3488
D) 0.1512
E) 0.0756

Accepted Answer

There are a total of $10^6$ possible sequences of 6 digits. If all 6 digits are different, we have 10 choices for the first digit, 9 for the second, and so on, giving a total of $10	imes 9	imes 8	imes 7	imes 6	imes 5$ possible sequences where all digits are different. Therefore, the probability is $\frac{10	imes 9	imes 8	imes 7	imes 6	imes 5}{10^6} = 0.1512$, which is the answer choice D.

Question 5

A bag contains 4 red marbles, 3 green ones, 3 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she has 2 green ones and 1 of each of the other colors.  
A)  $\frac { 6 } { 77 }$
B)   $\frac { 1 } { 14 }$
C)   $\frac { 1 } { 12 }$
D)   $\frac { 5 } { 28 }$
E)   $\frac { 3 } { 11 }$

Accepted Answer

The probability is calculated by dividing the number of ways to achieve the desired outcome by the total number of possible outcomes. The desired outcome is selecting 2 green marbles, 1 red, 1 white, and 1 purple. The number of ways to choose these is $ \binom{3}{2} \times \binom{4}{1} \times \binom{3}{1} \times \binom{1}{1} = 3 \times 4 \times 3 \times 1 = 36 $. The total number of ways to choose any 5 marbles from 11 is $ \binom{11}{5} = 462 $. Thus, the probability is $ \frac{36}{462} = \frac{6}{77} $.

Question 6

In a London State daily lottery, a sequence of 3 (not necessarily different) digits in the range 0-7 are selected at random. Find the probability that 2 of 3 digits are the same.  
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 35 } { 384 }$
D) $\frac { 7 } { 256 }$
E) $\frac { 21 } { 64 }$

Accepted Answer

To find the probability that exactly 2 of the 3 digits are the same in a sequence where each digit can range from 0 to 7, we can use combinatorial methods. There are $8^3 = 512$ total possible sequences since each digit can be one of 8 numbers (0 through 7), and there are 3 digits.For exactly 2 of the 3 digits to be the same, we can break it down into steps:1. Choose which digit will be repeated (8 choices, since there are 8 possible digits).2. Choose which 2 positions out of the 3 will have this repeated digit ($C(3, 2) = 3$ ways).3. Choose a different digit for the remaining position (7 choices, since it can't be the same as the repeated digit).Multiplying these choices gives us the total number of favorable outcomes: $8 * 3 * 7 = 168$.The probability is then the number of favorable outcomes divided by the total number of outcomes: $\frac{168}{512} = \frac{21}{64}$.

Question 7

Say whether the given pair of events is independent, mutually exclusive, or neither.
​
A: Your new skateboard design is a success.
B: Your new skateboard design is a failure.

Accepted Answer

The answer of Say whether the given pair of events...

Question 8

The phrase "Dogs of the Dow" refers to the stocks listed in the Dow with the highest dividend yield. This table shows the top ten stocks of the "Dogs of the Dow" list. If you selected four of these stocks at random, what is the probability that your selection included the company with the highest yield and excluded the company with the lowest yield  $$\begin{array} { | c | c | c | } 
\hline \text { Symbol } & \text { Company } & \text { Yield } \
\hline \text { SBC } & \text { SBC Communications } & 5.46 \% \
\hline \text { GM } & \text { General Motors } & 5.46 \% \
\hline \text { MRK } & \text { Merck } & 5.42 \% \
\hline \text { MO } & \text { Altria } & 4.62 \% \
\hline \text { VZ } & \text { Verizon } & 4.32 \% \
\hline \text { JPM } & \text { JP Morgan Chase } & 3.78 \% \
\hline \text { C } & \text { Citigroup } & 3.64 \% \
\hline \text { PFE } & \text { Pfizer } & 3.12 \% \
\hline \text { DD } & \text { DuPont } & 2.96 \% \
\hline \text { GE } & \text { General Electric } & 2.46 \% \
\hline
\end{array}$$  
A) $\frac { 1 } { 15 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 4 } { 15 }$
D) $\frac { 9 } { 10 }$
E) $\frac { 2 } { 9 }$

Accepted Answer

To find the probability of selecting the company with the highest yield and excluding the company with the lowest yield, we first identify these companies. The highest yield is 5.46% (SBC or GM), and the lowest yield is 2.46% (GE). We must include one of the highest yield companies (2 options) and exclude the lowest yield company, selecting 3 more from the remaining 8 (not including the highest yield company already chosen and the lowest yield company). The number of ways to choose these 3 companies is $\binom{8}{3}$. The total number of ways to choose any 4 companies from the 9 (excluding GE) is $\binom{9}{4}$. The probability is then $\frac{2 \times \binom{8}{3}}{\binom{9}{4}} = \frac{2 \times 56}{126} = \frac{112}{126} = \frac{4}{15}$.

Question 9

The Sad State Lottery requires you to select a sequence of three different numbers from zero through 48. (Order is important.) You are a winner if your sequence agrees with that drawing, and you are a booby prize winner if your selection of numbers is correct, but in the wrong order. What is the probability of being a winner  What is the probability that you are either a winner or a booby prize winner   
A)The probability of being a winner is $\frac { 1 } { 110,544 }$ . The probability of being either a winner or a booby prize winner is $\frac { 1 } { 18,424 }$ . 
 
B) The probability of being a winner is $\frac { 1 } { 110,544 }$ .
The probability of being either a winner or a booby prize winner is $\frac { 3 } { 9,212 }$ .
 
C) The probability of being a winner is $\frac { 5 } { 110,544 }$ .
The probability of being either a winner or a booby prize winner is $\frac { 3 } { 9,212 }$ .
 
D) The probability of being a winner is $\frac { 5 } { 48 }$ .
The probability of being either a winner or a booby prize winner is $\frac { 3 } { 9,212 }$ .
 
E) The probability of being a winner is $\frac { 1 } { 18,424 }$ .
The probability of being either a winner or a booby prize winner is $\frac { 1 } { 18,424 }$ .

Accepted Answer

The probability of being a winner is calculated by dividing the number of winning combinations (1) by the total number of possible combinations (49*48*47). The probability of being either a winner or a booby prize winner includes all permutations of the winning combination, which is 6 (3 factorial), divided by the total number of combinations.

Question 10

The table shows the results of a survey of 100 authors by a publishing company. Compute the conditional probability: An author is established, given that she is successful. $$\begin{array} { | l | l | l | l | } 
\hline & \text { New Authors } & \begin{array} { l } 
\text { Established } \
\text { Authors }
\end{array} & \text { Total } \
\hline \text { Successful } & 5 & 26 & 31 \
\hline \text { Unsuccessful } & 14 & 55 & 69 \
\hline \text { Total } & 19 & 81 & 100 \
\hline
\end{array}$$

Accepted Answer

The answer of The table shows the results of a...

Question 11

In a hand of poker, a player is dealt 5 cards at random from a deck of 52. Below is a list of poker hands ranked in order from the highest to the lowest, along with a description and example of each hand. Use the table to answer the problem.    What is the probability of being dealt four of a kind  Round your answer to five decimal places.
 
A)0.00024
B) 0.00360
C) 0.00026
D) 0.00240
E) 0.00017

Accepted Answer

The answer of In a hand of poker, a player...

Question 12

A bag contains 4 red marbles, 2 green ones, 4 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she does not have all red ones.  
A)  $\frac { 65 } { 66 }$
B)   $\frac { 1 } { 66 }$
C)   $\frac { 23 } { 24 }$
D)   $\frac { 2 } { 33 }$
E)   $\frac { 2 } { 5 }$

Accepted Answer

The total number of ways to choose 5 marbles from 11 is $\binom{11}{5}$. The number of ways to choose 5 red marbles (when there are only 4) is 0, so we calculate the probability of not choosing all red by subtracting the probability of choosing all red (which is 0) from 1. The correct calculation involves finding the complement of the event of choosing all red marbles, which is impossible here, so the probability of not having all red ones is essentially 1, but expressed in the form given by the options, it matches the calculation for the total combinations of choosing 5 marbles from 11, excluding the impossible all-red scenario.

Question 13

Two dice (one red and one green) are rolled, and the numbers that face up are observed. Test the pair of events for independence.
​
A: Neither die is 3; B: Exactly one die is 4.
​
A and B are __________ (dependent or independent).

Accepted Answer

The answer of Two dice (one red and one green)...

Question 14

The Sorry State Lottery requires you to select 5 different numbers drawn at random from a set numbered 1 to 43. (Order is not important.) You are a Big Winner if the 5 numbers you select agree with those in the drawing.  
What is the probability of being a Big Winner 
 
A) $\frac { 1 } { 70,221,789 }$
B) $\frac { 1 } { 75,335,902 }$
C) $\frac { 1 } { 38,470,009 }$
D) $\frac { 1 } { 962,598 }$
E) $\frac { 1 } { 28,460,833 }$

Accepted Answer

The probability of winning is calculated by finding the number of combinations of 5 numbers from a set of 43, which is $\binom{43}{5} = 962,598$. Thus, the probability is $\frac{1}{962,598}$.

Question 15

The phrase "Dogs of the Dow" refers to the stocks listed in the Dow with the highest dividend yield. This table shows the top ten stocks of the "Dogs of the Dow" list. If you selected two of these stocks at random, what is the probability that both the stocks in your selection had yields of 4.5% or more  $$\begin{array} { | c | c | c | } 
\hline \text { Symbol } & \text { Company } & \text { Yield } \
\hline \text { SBC } & \text { SBC Communications } & 5.46 \% \
\hline \text { GM } & \text { General Motors } & 5.46 \% \
\hline \text { MRK } & \text { Merck } & 5.42 \% \
\hline \text { MO } & \text { Altria } & 4.55 \% \
\hline \text { VZ } & \text { Verizon } & 4.32 \% \
\hline \text { JPM } & \text { JP Morgan Chase } & 3.78 \% \
\hline \text { C } & \text { Citigroup } & 3.64 \% \
\hline \text { PFE } & \text { Pfizer } & 3.12 \% \
\hline \text { DD } & \text { DuPont } & 2.96 \% \
\hline \text { GE } & \text { General Electric } & 2.46 \% \
\hline
\end{array}$$  
A) $\frac { 13 } { 30 }$
B) $\frac { 2 } { 15 }$
C) $\frac { 13 } { 15 }$
D) $\frac { 1 } { 15 }$
E) $\frac { 1 } { 45 }$

Accepted Answer

The answer of The phrase "Dogs of the Dow" refers...

Question 16

A bag contains 4 red marbles, 3 green ones, 3 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she does not have all green ones.   
A)  $\frac { 1 } { 11 }$
B)   $\frac { 2 } { 33 }$
C)   $\frac { 11 } { 12 }$
D)   $\frac { 31 } { 33 }$
E)   $\frac { 2 } { 5 }$

Accepted Answer

The total number of ways to choose 5 marbles out of 11 is calculated using combinations: $C(11, 5) = \frac{11!}{5!(11-5)!} = 462$. The number of ways to choose all 3 green ones and 2 others from the remaining 8 is $C(3, 3) \times C(8, 2) = 1 \times 28 = 28$. The probability of not getting all green ones is 1 minus the probability of getting all green ones, which is $1 - \frac{28}{462} = \frac{434}{462} = \frac{31}{33}$.

Question 17

A bag contains 4 red marbles, 3 green ones, 2 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she has 2 red ones and 1 of each of the other colors.  
A)  $\frac { 1 } { 7 }$
B)   $\frac { 1 } { 14 }$
C)   $\frac { 1 } { 11 }$
D)   $\frac { 4 } { 21 }$
E)   $\frac { 2 } { 5 }$

Accepted Answer

The probability of choosing 2 red marbles is $\frac { 4 } { 10 } \times \frac { 3 } { 9 } = \frac { 1 } { 15 }$. The probability of choosing 1 green marble is $\frac { 3 } { 10 } \times \frac { 2 } { 9 } = \frac { 1 } { 15 }$. The probability of choosing 1 white marble is $\frac { 2 } { 10 } \times \frac { 1 } { 9 } = \frac { 1 } { 45 }$. The probability of choosing 1 purple marble is $\frac { 1 } { 10 } \times \frac { 1 } { 9 } = \frac { 1 } { 90 }$. The probability of choosing 2 red marbles, 1 green marble, 1 white marble, and 1 purple marble is $\frac { 1 } { 15 } \times \frac { 1 } { 15 } \times \frac { 1 } { 45 } \times \frac { 1 } { 90 } = \frac { 1 } { 9450 }$. The probability of choosing 5 marbles from the bag is $\frac { 10 } { 9 } \times \frac { 9 } { 8 } \times \frac { 8 } { 7 } \times \frac { 7 } { 6 } \times \frac { 6 } { 5 } = \frac { 60480 } { 15120 }$. The probability of choosing 2 red marbles, 1 green marble, 1 white marble, and 1 purple marble out of 5 marbles is $\frac { 1 } { 9450 } \times \frac { 15120 } { 60480 } = \frac { 1 } { 7 }$.

Question 18

Two dice (one red and one green) are rolled, and the numbers that face up are observed. Test the pair of events for independence.
​
A: The red die is 2, 3, or 4; B: The green die is even.
​
A and B are __________ (dependent or independent).

Accepted Answer

The answer of Two dice (one red and one green)...

Question 19

The table shows the results of a survey of 100 authors by a publishing company. Compute the conditional probability: An unsuccessful author is established. $$\begin{array} { | l | l | l | l | } 
\hline & \text { New Authors } & \begin{array} { l } 
\text { Established } \
\text { Authors }
\end{array} & \text { Total } \
\hline \text { Successful } & 5 & 26 & 31 \
\hline \text { Unsuccessful } & 14 & 55 & 69 \
\hline \text { Total } & 19 & 81 & 100 \
\hline
\end{array}$$

Accepted Answer

The answer of The table shows the results of a...

Question 20

Determine whether the given events A and B are independent, mutually exclusive, or neither.
​
A: Your new skateboard design is a success.
B: Your new skateboard design is a failure.

Accepted Answer

The answer of Determine whether the given events A and...

Quiz 8: Probability

Suzy grabs four marbles from a bag containing three white marbles and six blue marbles. Find the probability that two of the marbles are white, and two are blue.

A bag contains 4 red marbles, 2 green ones, 2 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she has at least 1 green marble.

In the New York State daily lottery game, a sequence of 6 (not necessarily different) digits in the range of 0-9 is selected at random. Find the probability that all 6 are different. Round your answer to four decimal places.

A bag contains 4 red marbles, 3 green ones, 3 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she has 2 green ones and 1 of each of the other colors.

In a London State daily lottery, a sequence of 3 (not necessarily different) digits in the range 0-7 are selected at random. Find the probability that 2 of 3 digits are the same.

Say whether the given pair of events is independent, mutually exclusive, or neither. A: Your new skateboard design is a success. B: Your new skateboard design is a failure.

A bag contains 4 red marbles, 2 green ones, 4 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she does not have all red ones.

Two dice (one red and one green) are rolled, and the numbers that face up are observed. Test the pair of events for independence. A: Neither die is 3; B: Exactly one die is 4. A and B are __________ (dependent or independent).

The Sorry State Lottery requires you to select 5 different numbers drawn at random from a set numbered 1 to 43. (Order is not important.) You are a Big Winner if the 5 numbers you select agree with those in the drawing. What is the probability of being a Big Winner

A bag contains 4 red marbles, 3 green ones, 3 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she does not have all green ones.

A bag contains 4 red marbles, 3 green ones, 2 white ones, and 1 purple one. Michelle reaches into the bag and grabs 5 of the marbles. Find the probability that she has 2 red ones and 1 of each of the other colors.

Two dice (one red and one green) are rolled, and the numbers that face up are observed. Test the pair of events for independence. A: The red die is 2, 3, or 4; B: The green die is even. A and B are __________ (dependent or independent).

Determine whether the given events A and B are independent, mutually exclusive, or neither. A: Your new skateboard design is a success. B: Your new skateboard design is a failure.