The general manager, marketing director, and 3 other employees of Company A are hosting a visit by the vice president and 2 other employees of Company B. The eight people line up in a random order to take a photo. Every way of lining up the people is equally likely.
(a) What is the probability that the general manager is next to the vice president?
(b) What is the probability that the marketing director is in the leftmost position?
(c) Determine whether the two events are independent. Prove your answer by showing that one of the conditions for independence is either true or false.

Solution :

Let the three places be 1, 2, 3, 4, 5, 6, 7, 8

a). Number of the cases when a general manager is the next to a vice president is equal to 7 and the these 2 can be arranged in 21 ways. So the total number of ways = 7 x 2

                  = 14

[(1,2)(2,1) (2,3)(3,2) (3,4)(4,3) (4,5)(5,4) (5,6)(6,5) (6,7)(7,8) (8,7)(7,6)]

Therefore the required probability is

  [tex]$=\frac{14}{8!}$[/tex]

 = [tex]$\frac{14}{40320} = 0.000347$[/tex]

b). The probability that the marketing director to be placed in the leftmost position is

   [tex]$=\frac{7!}{8!}$[/tex]

  [tex]$=\frac{1}{8} = 0.125$[/tex]

c). The two events are not independent because

   [tex]$P(A \cap B) \neq P(A) \times P(B)$[/tex]

  [tex]$\frac{12}{8!} \neq \frac{14}{8!} \times \frac{1}{8}$[/tex]

where A is the case a and B is the case b.

(a) The possibility of the general manager is next to the vice president is [tex]\frac{1}{4}[/tex].

(b) The possibility of the marketing director is in the leftmost position is [tex]\frac{1}{8}[/tex].

(c) So, the two events are dependent on each other.

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. The probability of all the events in a sample space adds up to 1.

Total people in company A and company B is [tex]=8[/tex]

Overall ways in which these [tex]8[/tex] people can be lined up[tex]=8![/tex]

                                                                                        [tex]=40320[/tex]

(a) The probability that the general manager is next to the vice president is[tex]=P(A)[/tex]

Now, we can combine the general manager and vice president as one, then the total people in both the company will become [tex]7[/tex].

by arranging these [tex]7[/tex] people in one line [tex]=7![/tex]

                                                                [tex]=5040[/tex]

Again, combine the general manager and vice president in one line[tex]=2![/tex]

                                                                                                                [tex]=2[/tex]

Therefore, [tex]P(A)=\frac{5040\times 2}{40320}[/tex]

                          [tex]=\frac{10080}{40320}[/tex]

[tex]P(A)=\frac{1}{4}[/tex]

(b) The probability that the marketing director is in the leftmost position is[tex]=P(B)[/tex]

Now, fixing the position of marketing director in the leftmost.

arranging the [tex]7[/tex] other people in [tex]7![/tex] ways [tex]=5040[/tex]

Therefore,[tex]P(B)=\frac{5040}{40320}[/tex]

                          [tex]=\frac{1}{8}[/tex]

[tex]P(B)=\frac{1}{8}[/tex]

(c) Assuming event B already occurred which means that the position of marketing director is already fixed in the leftmost position.

Now, trying to find out the probability of the general manager next to the vice president is event A. it comes different because we are not allowed to arrange rest [tex]7[/tex] people, we have to fix the position of one person that causes the repetition of probability.

So, the two events are dependent on each other.

Learn more about the topic of Probability: https://brainly.com/question/26959834

HELP ASAPPPP I WILL GIVE BRAINLIEST

1/4 x 7/8 in simplest form please. what’s 1/6 x 5/6 in simplest form too.

Which distribution is most nearly appropriate to describe_______the number of fatalities in Texas in a given year due to poisonous snakebites.

Please help (8th grade math)

Solve the equation. w/5 =3 w=​

HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!HELP!​

Caroline is saving up to buy a new video game. She already has $45 and can save an additional $9 per week using money from her after school job. How much total money would Caroline have after 9 weeks of saving? Also, write an expression that represents the amount of money Caroline would have saved in w weeks

2) Addisyn has a 12-sided die. If she rolls
the die 250 times, how many times can
Addisyn predict she will roll a 9?
A. about 9 times
B. about 27 times
C. about 21 times
D. about 108 times

The circumference of a circle that has a radius of 70 feet is most nearly

A. 220 feet.

B. 440 feet.

C. 660 feet.

D. 690 feet.​

38) Find the slope of the line containing the points (2, 7) and (-5, -4).
A. 711
B. 117
C. −511
D. -1

Help here is an easy one

Ivy and Andrey were asked to find an explicit formula for the sequence -100,-50,0,50,...−100,−50,0,50,...minus, 100, comma, minus, 50, comma, 0, comma, 50, comma, point, point, point, where the first term should be f(1)f(1)f, left parenthesis, 1, right parenthesis.
Ivy said the formula is f(n)=-100+50(n-1)f(n)=−100+50(n−1)f, left parenthesis, n, right parenthesis, equals, minus, 100, plus, 50, left parenthesis, n, minus, 1, right parenthesis.
Andrey said the formula is f(n)=-150+50nf(n)=−150+50nf, left parenthesis, n, right parenthesis, equals, minus, 150, plus, 50, n.
Which one of them is right?

express the surface area A of a cube in terms of its volume V

If an angle not acute straight or obtuse it must be a?

Write a slope-intercept equation of the line with a slope of — that passes through the point (6, 10).
O A. y = 6x + 10
O B. y=-x +38 3
O C. y=-x +14
O d.y= -x + 6

34 1/3% of animals at the shelter are dogs. About what fraction of the animals are dogs?

I will give a brainlist if someone could answer this problem with a explanation:)

Trignometery: Sin Cos and Tan

Two linear functions are shown below which function has the greater rate of change? Use the drop-down menus to show your answer
1. A or B
2. A or B
3. A or B

Lisa bought bottles of soda for $2.50 each and a large pizza for $17. The total cost was $27. How many bottles of soda did Lisa buy?

A.
4

B.
3

C.
10

D.
6

Which brand would Cala need more of to fertilize her 450-foot row of vegetables

8x-5y=37 x=2y+6 substitution

Expand the logarithm. Simplify, if possible.
log7(49^w)

Choose the best estimate for the division problem below. 7.9 0.8 O A. 10 B. 8 O C. 15​

Please answer the question attached, marking brainliest (serious answers only)

I need help with this question​

PLEASE HELP AND LEAVE A SMALL EXPLANATION

Kendra just bought a new house and needs to buy new sod for her backyard. If the dimensions of her yard are 24.6 feet by 14.8 feet , what is the area of her yard

What is a factor that is less than 1?

You work for a landscaping company and will be installing a fence around a customer’s yard. The yard is rectangular and is 45 feet wide by 80 feet long. About how many feet of fencing should you buy to enclose the entire yard?