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Probability Concept Scenarios

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Please help me to check my answers to the following questions. Please show your work so that I may follow and compare against my own:

a) In how many ways can seven banquet speakers be seated along one side of the head table?

b) Gym lockers are to be numbered from 1 to 99 using individual metal number plates to be placed on each locker. How many 7's are needed?

c) In how many ways can a hostess place six name placecards around a round table?

d) In how many ways can five different keys be put in a flat leather key case?

e) In how many ways can five different keys be put on a key ring?

f) You have one apple, one orange, one banana, and one grapefruit. How many different ways can you hand out all the fruit to 5 people, if no person gets more than one kind of fruit?

g) If I am to paint our house by myself, it will take 4 days to complete painting the house. If my wife is to paint the house alone, it will take 6 days to complete her work. How many days will it take to paint our house if we decide to work together (without fight)?

h) Joey and Ross along with 4 other best friends go to see a movie. They find a row of 6 seats, but Joey and Ross don't want to sit next each other. How many different seating arrangements are possible if Joey and Ross don't want to sit next each other?
?

[2](10) At a certain gas station, 40% of all customers fill their tanks. Of those who fill their tanks, 80% pay with a credit card.
a) What is the probability that a randomly selected customer fills his or her tank and pays with a credit card?

b) If three customers are randomly selected, what is the probability that all three fill their tanks and pay with a credit card?

?
[3](10) There is a saying about initial public offerings (IPOs) of stock: "If you want it, you can't get it; if you can get it, you don't want it." This is because it is often difficult for the general public to obtain shares initially when a "hot" new company first goes on sale. Instead, most of us have to wait until it starts trading on the open market, often at a substantially higher price. Suppose that, given that you can obtain shares at the initial offering, the probability of the stock performing well is 0.35. However, given that you are unable to initially purchase shares, the probability of the stock performing well is 0.80. Overall, assume that you can obtain shares in about 15% of IPOs.
a) Find the probability that both you are able to purchase the stock at the initial offering and the stock performs well.

b) Find the probability that the stock turned out not to perform well if you were unable to obtain such shares.

c) How much access to successful IPOs do you have? That is, what is the probability that you can purchase successful IPOs?

d) What percentage of the time, over the long run, will you be pleased with the outcome?

[4](10) You are to select two cards one at a time from a well-shuffled deck of 52 playing cards (jokers not allowed). You want to get a heart as the first card and a King (K) as the second.
a) If you are given one of the following two options to do this, which option would you choose,
? Option #1: to be allowed to select the first card and put it back into the deck before selecting the second card, or
? Option #2: to be allowed to keep the first card (not to put it back to the deck) and select the second one?
Explain why you think your choice is better than the other option.

b) What is the probability to select such two cards under option #1?

c) What is the probability to select such two cards under option #2?

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The probability concept scenarios are examined.

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[1]
a) In how many ways can seven banquet speakers be seated along one side of the head table?
Solution: 7!=7∙6∙5∙4∙3∙2∙1=5040 ways

b) Gym lockers are to be numbered from 1 to 99 using individual metal number plates to be placed on each locker. How many 7's are needed?
Solution: The plates containing 7's are
7, 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 87, 97
So 20 7's are needed.

c) In how many ways can a hostess place six name place cards around a round table?
Solution: The number of ways to seat n people at a round table is (n-1)!
So there are 5!=120 different ways in which a hostess can place six name place cards around a round table.

d) In how many ways can five different keys be put in a flat leather key case?
Solution: 5!=120 ways.

e) In how many ways can five different keys be put on a key ring?
Solution: Since the keys are arranged in a circle, there are 4!=24 different way to put the keys on a key ring.

f) You have one apple, one orange, one banana, and one grapefruit. How many different ways can you hand out all the fruit to 5 people, if no person gets more than one kind of fruit?
Solution: We have 5 choices for the first person (apple, orange, banana, grapefruit, or no fruit), then 4 choices for the second, then 3 choices for the third, then 2 choices for the fourth, and then finally 1 choice for the fifth. So there are 5!=120 ways to hand out all the fruit.

g) If I am to paint our house by myself, it will take 4 days to complete painting the house. If my wife is to paint the house alone, it will take 6 days to complete her work. How many days will it take to ...

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