Question 1

Two events A and B are if the fact that A occurs does not 128)
affect the probability of B occurring.

Accepted Answer

The answer of Two events A and B are if...

Question 2

$$\text { Evaluate the permutation: } { } _ { 10 } P _ { 8 }$$

Accepted Answer

The permutation formula is $nPr = \frac{n!}{(n-r)!}$. Here, $n = 10$ and $r = 8$, so $ _{10}P_{8} = \frac{10!}{(10-8)!} = \frac{10!}{2!} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{2} = 1,814,400$.

Question 3

In a second grade class containing 15 girls and 11 boys, 2 students are selected at random to give out the math papers. What is the probability that the second student chosen is a
Boy, given that the first one was a girl?

Accepted Answer

After the first girl is chosen, there are 25 students left (15-1 girls + 11 boys = 25 students). The probability that the next student chosen is a boy is 11 out of these 25, which simplifies to $\frac{11}{25}$.

Question 4

On an eight-question true-false quiz, a student guesses each answer. What is the
probability that the student gets at least one of the answers correct?

Accepted Answer

The answer of On an eight-question true-false quiz, a student...

Question 5

The probability of an event B in relationship to an event A is
defined as the probability that event B occurs after event A has already occurred.

Accepted Answer

The answer of The probability of an event B in...

Question 6

There are 3 first grade children, 5 second grade children, and 7 third grade children in a room. When choosing two children, what is the conditional probability of choosing a
Second grade child, given that either a first or a third grade child was chosen first?

Accepted Answer

Given that either a first or third grade child was chosen first, we have 10 children left (3+7=10, excluding second graders initially). The probability of choosing a second grade child next is 5 out of the remaining 14 children (5 second graders + 9 first and third graders), so $\frac{5}{14}$.

Question 7

A box contains blue chips and red chips. A person selects two chips without replacement. If the probability of selecting a blue chip and a red chip is $\frac { 1 } { 4 }$, and the probability of selecting a blue chip on the first draw is $\frac { 5 } { 16 }$, find the probability of selecting the red chip on the second draw, given that the first chip selected was a blue chip.

Accepted Answer

The answer of A box contains blue chips and red...

Question 8

When the outcome of event A does not affect the probability of the outcome of a
subsequent event B, then the events are said to be dependent.

Accepted Answer

The answer of When the outcome of event A does...

Question 9

If the letters A, B, C, D, E, and F are to be used in a five-letter code, how many different codes are possible if repetitions are not permitted?

Accepted Answer

Since repetitions are not allowed and there are 6 different letters to choose from for the first position, 5 for the second, 4 for the third, 3 for the fourth, and 2 for the fifth, the total number of different codes is calculated by multiplying these numbers together: 6 * 5 * 4 * 3 * 2 = 720.

Question 10

When two events are independent, the probability of both occurring is:

Accepted Answer

The probability of two independent events both occurring is found by multiplying their individual probabilities, as stated in choice A.

Question 11

Below are listed the numbers of engineers in various fields by sex. Choose one engineer at random. Find $P$ (electrical|male).
$\begin{array} { l c c c } & \text { Mechanical } & \text { Electrical } & \text { Biomedical } \ \text { Male } & 9691 & 4997 & 6064 \ \text { Female } & 2061 & 1015 & 5937 \end{array}$

Accepted Answer

$P(\text{electrical|male}) = \frac{\text{Number of male electrical engineers}}{\text{Total number of male engineers}} = \frac{4997}{9691+4997+6064} = \frac{4997}{20752} \approx 0.241.$

Question 12

There are 3 blue balls, 5 red balls, and 2 white balls in a bag of balls. If a person selects two of the balls, what is the probability that the second one is blue given that the first one
Was white?

Accepted Answer

After removing a white ball, there are 9 balls left (3 blue, 5 red, 1 white). The probability of selecting a blue ball next is 3 out of 9, which simplifies to $\frac{1}{3}$.

Question 13

How many different ways can 5 brands of soda be displayed on a supermarket's shelf, one at a time in a row?

Accepted Answer

This is a permutation problem because the order in which the soda brands are displayed matters. The number of ways to arrange 5 brands is 5! (5 factorial), which is 5 x 4 x 3 x 2 x 1 = 120.

Question 14

A gumball machine contains 300 grape flavored balls, 400 cherry flavored balls, and 500 lemon flavored balls. What is the probability of getting 1 grape ball, 1 cherry ball, and 1
Lemon ball if each ball was removed and then replaced before choosing the next from the
Machine?

Accepted Answer

The answer of A gumball machine contains 300 grape flavored...

Question 15

$${ } _ { 7 } ^ { P } 6 \text { is equal to } 7 \text {. }$$

Accepted Answer

${ } _ { 7 } ^ { P } 6$ represents the number of permutations of 7 items taken 6 at a time, which is calculated as $7! / (7-6)!$, equal to $7 \times 6 \times 5 \times 4 \times 3 \times 2 = 5040$, not 7.

Question 16

Evaluate the factorial: 10!

Accepted Answer

The factorial of 10 (denoted as 10!) is the product of all positive integers from 1 to 10, which equals 3,628,800.

Question 17

In a physics class there are 12 juniors and 16 males; so there must be 28 male juniors in
this class.

Accepted Answer

The statement assumes that all juniors and males are distinct groups without overlap, but some of the juniors could be males, so the total number of male juniors cannot be determined from the information given.

Question 18

A store manager wants to display 4 different brands of toothpaste in a row. How many ways can this be done?

Accepted Answer

The number of ways to arrange 4 different brands of toothpaste in a row is calculated using the formula for permutations of n distinct objects, which is n!, where n is the number of objects. Here, n=4, so the calculation is 4! = 4*3*2*1 = 24.

Question 19

In the Happy Hilltop Health Home, 10% of the residents play shuffleboard, 35% of the residents play poker, and 15% of the residents garden. If 5% of the residents play poker
And garden, find the probability that a resident plays poker, given that they also garden.

Accepted Answer

The probability that a resident plays poker given that they also garden is found using the formula for conditional probability: P(A|B) = P(A and B) / P(B). Here, A is the event of playing poker, and B is the event of gardening. Given that 5% of the residents play poker and garden (P(A and B) = 5%), and 15% of the residents garden (P(B) = 15%), the probability is P(A|B) = 5% / 15% = 33.3%.

Question 20

$$\text { Evaluate the combination: } { } _ { 12 } C _ { 8 }$$

Accepted Answer

${ }_{12}C_{8} = \frac{12!}{8!(12-8)!} = \frac{12!}{8!4!} = 495$. This is calculated using the combination formula, which counts the number of ways to choose 8 items from 12, without regard to order.

Quiz 4: Probability and Counting Rules

Two events A and B are if the fact that A occurs does not 128) affect the probability of B occurring.

Evaluate the permutation: 10P8\text { Evaluate the permutation: } { } _ { 10 } P _ { 8 } Evaluate the permutation: 10​P8​

In a second grade class containing 15 girls and 11 boys, 2 students are selected at random to give out the math papers. What is the probability that the second student chosen is a Boy, given that the first one was a girl?

On an eight-question true-false quiz, a student guesses each answer. What is the probability that the student gets at least one of the answers correct?

The probability of an event B in relationship to an event A is defined as the probability that event B occurs after event A has already occurred.

There are 3 first grade children, 5 second grade children, and 7 third grade children in a room. When choosing two children, what is the conditional probability of choosing a Second grade child, given that either a first or a third grade child was chosen first?

When the outcome of event A does not affect the probability of the outcome of a subsequent event B, then the events are said to be dependent.

If the letters A, B, C, D, E, and F are to be used in a five-letter code, how many different codes are possible if repetitions are not permitted?

When two events are independent, the probability of both occurring is:

There are 3 blue balls, 5 red balls, and 2 white balls in a bag of balls. If a person selects two of the balls, what is the probability that the second one is blue given that the first one Was white?

How many different ways can 5 brands of soda be displayed on a supermarket's shelf, one at a time in a row?

A gumball machine contains 300 grape flavored balls, 400 cherry flavored balls, and 500 lemon flavored balls. What is the probability of getting 1 grape ball, 1 cherry ball, and 1 Lemon ball if each ball was removed and then replaced before choosing the next from the Machine?

7P6 is equal to 7. { } _ { 7 } ^ { P } 6 \text { is equal to } 7 \text {. }7P​6 is equal to 7.

Evaluate the factorial: 10!

In a physics class there are 12 juniors and 16 males; so there must be 28 male juniors in this class.

A store manager wants to display 4 different brands of toothpaste in a row. How many ways can this be done?

In the Happy Hilltop Health Home, 10% of the residents play shuffleboard, 35% of the residents play poker, and 15% of the residents garden. If 5% of the residents play poker And garden, find the probability that a resident plays poker, given that they also garden.

Evaluate the combination: 12C8\text { Evaluate the combination: } { } _ { 12 } C _ { 8 } Evaluate the combination: 12​C8​

$\text { Evaluate the permutation: } { } _ { 10 } P _ { 8 }$

${ } _ { 7 } ^ { P } 6 \text { is equal to } 7 \text {. }$

$\text { Evaluate the combination: } { } _ { 12 } C _ { 8 }$