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## Continuous Vs. Discrete: Ex. 1

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Continuous Vs. Discrete: Ex. 1

Supposed that you draw 3 cards from a deck of cards and record whether the suit is black or red. Let X be the total number of red cards.

- construct a table that shows the values of X

Identify the following events in words and as a subset of the sample space.

b) {X = 2} d) {X 2} e) {X 1}

Continuous Vs. Discrete: Ex. 4

The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

What is the probability that the train leaves before 8:06 or after 8:21?

Continuous Vs. Discrete: Ex. 2

The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

What is the CDF of X?

Continuous Vs. Discrete: Ex. 3

Suppose that you roll a 4-side die 3 times. Let X be the total number of '1's that are rolled.

Determine the CDF of X.

Example A

Suppose that 10% of engines manufacture on a certain assembly line are defective. If engines are randomly selected one at a time and tested, find the probability that exactly two defective engines will be tested before the a good engine is found.

Example B

A family decides to have children until it has three children of the same gender. Assuming that it is equally likely to have either a boy or a girl, what is the probability that the family will have 4 children?

Example C

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages. Consider a sample of 25 incoming calls, what is the probability that at most 6 of the calls involve a fax?

Example D

A personnel director interviewing 11 teachers for four job openings has scheduled six interviews for the first day and five for the second day of interviewing. Assume that the candidates are interviewed in random order. What is the probability that the four top candidates are interviewed on the first day?

Example E

When circuit boards used in the manufacture of compact disc players are tested, the long run percentage of defectives is 5%. When you have 25 circuit boards, what is the probability that the number of defective boards is at least 2?

Example F

A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis. What is the probability that all specimens of one of the two types of rock are selected for analysis?

Example G

Suppose that the number of tornadoes observed in a particular region during a 1-year period has a parameter of 8/year. What is the probability that this region has 5 tornadoes in the next year?

Example H

What is the probability that a 15 or higher is observed when a person rolls a 20-sided die?

Example I

A geological study indicates that an exploratory oil well drilled in a certain region should strike oil with a probability of 0.25. What is the probability that the first strike of oil comes after drilling three dry (nonproductive) wells?

Example J

Suppose that the number of drivers who travel between a particular origin and destination during one year has a parameter of 20/year. What is the probability that the number of drivers will exceed 20?

Example K

20% of the applicants for a certain sales position are fluent in English and Spanish. Supppose that four jobs requiring fluency in English and Spanish are open. Find the probability that six total people are interviewed before the fourth qualified applicant.

Example L

The number of people arriving for treatment at an emergency room can be modeled with a distribution that has a parameter of five/hour. What is the probability that 4 people will arrive in the next 45 minutes?

Example 1

Every second, 1.8 cosmic rays hit a specific spot on earth. Assume that we start counting at t = 0 seconds.

- What is the probability that there are exactly 12 cosmic rays hitting the spot between 10 seconds and 15 seconds?
- Given that exactly 3 cosmic rays hit the spot between 4 seconds and 5 seconds, what is the probability that 12 cosmic rays hit the spot between 10 seconds 15 seconds?
- What is the probability that at least one cosmic ray hits the spot between 4 seconds and 5 seconds and between 10 seconds and 15 seconds?

Example 2

The Lotto. In the Hoosier lotto, a player specifies six numbers of her choice from the numbers 1 – 48. In the lottery drawing, six winning numbers are chosen at random without replacement from the numbers 1 – 48. To win a prize, a lotto ticket must contain three or more of the winning numbers.

- Determine the PMF of the r.v. X, the number of winning numbers on the player’s ticket.
- If the player buys one Lotto ticket, determine the probability that she wins a prize (at least 3 numbers correct).
- If the player buys one Lotto ticket per week for a year, determine the probability that she wins a prize at least once in the 52 tries.

Example 3

A charitable organization is conducting a raffle in which the grand prize is a new car. Five thousand tickets, numbered 0001, 0002, …, 5000 are sold at $10 each. At the grand-prize drawing, one ticket stub will be selected at random from the 5000 ticket stubs. Let X denote the number on the ticket stub obtained.

- Find the PMF of the r.v. X.
- Suppose that you hold tickets numbered 1003 – 1025. Express the event that you win the grand prize in terms of the random variable X, and then compute the probability of the event.

Example 4

Between the hours of 2 and 4 pm, If the number of phone calls per minute coming into the switchboard of a company with parameter 2.5. Find the probability that the number of calls during one particular minute will be

- 0

b) 2 or fewer

Example 5

On my page of notes, I have 2150 characters. Say that the chance of a typo (after I proof it) is 0.001.

- What is the probability of exactly 1 typo on this page?
- What is the probability of at most 3 typos?

Example 6

Suppose that we roll an n-sided die until a '1' is rolled. Let X be the number of times it takes to roll the ninth '1'. What is the PMF of X?

Example 7

A textbook author is preparing an answer key for the answers in a book. In 500 problems, the author has made 25 errors. If a second person checks seven of these calculations randomly, what is the probability that he will detect two errors? Assume that the second person will definitely find the error in an incorrect answer.

Example 8

Suppose that we roll an n-sided die until a '1' is rolled. Let X be the number of times it takes to roll the '1'. What is the PMF of X?

Example 9

A restaurant serves eight entrées of fish, 12 of beef, and 10 of poultry. If customers select from these entrées randomly, what is the probability that

a) two of the next four customers order fish entrées?

b) at most one of the next four customers orders fish?

c) at least one of the next four customers orders fish?

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