Assignment (1)
Fall 2022
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Please follow the instructions below
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Write your name and Code number in the spaces provided.
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Give clear answers and explanations justifying your method wherever possible.
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Due Monday, Oct. 26, 2022, In the tutorial.
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Homework cannot be submitted as an e-mail attachment.
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Late submissions are not allowed.
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Question 1:
Roll two dice and multiply the numbers together.
(a) Write out the sample space.
(b) What is the probability that the product is a multiple of 6?
(c) What is the probability that the product is less than 10?
Question 2:
An MP3 player contains 60 songs: 30 rock, 15 jazz, and 15 blues. The player is set on random
play, so songs are played randomly and can be repeated. The duration of 20 rock songs, 10
jazz songs, and 8 blues songs is longer than 3 minutes. The duration of the other songs is less
than 3 minutes. We play four songs. Find the probability of each of the following events
(a) The four songs are all jazz;
(b) Among the four songs, none is rock;
(c) The first song is a rock song or takes longer than 3 minutes;
(d) If the first song took more than 3 minutes, what is the probability it was a rock song?
Question 3:
A bag contains 3 white balls, 4 green balls, and 5 red balls. Three balls are drawn from the
bag without replacement.
(a) Find the probability of selecting three different colors.
(b) Find the probability that the balls are all the same color.
Question 4:
A father, mother, 2 boys and 3 girls are asked to line up for a photograph. Determine the
number of ways they can line up if
(a) There are no restrictions;
(b) The parents stand together.
Question 5:
(a) How many distinct permutations can be made from the letters of the word 'bookkeeper'?
(b) How many different ten-digit numerals can be written using all of the following ten
digits: 1, 4, 3, 4, 3, 8, 4, 3, 1, 3
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Question 6:
A standard deck of 52 cards contains 4 aces. Suppose we choose a random ordering. Compute
the following
(a) The probability that all of the top 4 cards in the deck are aces;
(b) The probability that none of the top 4 cards in the deck is an ace.
Question 7:
A local postal carrier distributes first-class letters, advertisements, and magazines. For a
particular day, she spread the following numbers of each type of item:
Delivered to First-class letters Ads Magazines
Home
325
406
203
Business
732
1021
97
If an item of mail is selected at random, find these probabilities:
(a) The item went to a home;
(b) The item was an ad, or it went to a business;
(c) The item was a first-class letter, or it went to a home.
Question 8:
An insurance sales representative selects 3 policies to review. The group of policies he can
choose from contains 8 life policies, 5 automobile policies, and 2 homeowner policies. Find
the probability of selecting
(a) All life policies;
(b) Two are homeowner policies;
(c) All automobile policies;
(d) 1 of each policy;
(e) 2 life policies and 1 automobile policy.
Question 9:
A computer password consists of eight characters (repeating characters in passwords is
allowed).
(a) How many different passwords are possible if each character may be any lowercase letter
or digit?
(b) How many different passwords are possible if each character may be any lowercase letter
or digit, and at least one character must be a digit?
(c) A computer system requires that passwords contain at least one digit. Suppose eight
characters are generated at random, and each is equally likely to be any of the 26 letters
or 10 digits. What is the probability that a valid password will be generated?
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Question 10:
It is known from experience that in a particular industry, 60 percent of all labor-management
disputes are over wages, 15 percent are over working conditions, and 25 percent are over
fringe issues. Also, 45 percent of the disputes over wages are resolved without strikes, 70
percent of the disputes over working conditions are resolved without strikes, and 40 percent
of the disputes over fringe issues are resolved without strikes.
(a) What is the probability that a labor-management dispute in this industry will be resolved
without a strike?
(b) If a labor-management dispute in this industry is resolved without a strike, what is the
probability it was over wages?
Question 11:
While watching a game of Champions League football in a cafe in Manchester, you observe
someone supporting Manchester United. What is the probability that he was born within 25
miles of Manchester? Assume that:
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The probability that a randomly selected person in a typical local cafe environment is
born within 25 miles of Manchester is 1/20
The chance that a person born within 25 miles of Manchester supports United is 7/10,
and
The probability that a person not born within 25 miles of Manchester supports united
with a probability of 1/10.
Question 12:
A device has a sensor connected to an alarm system. The sensor triggers with a probability
of 0.95 if dangerous conditions exist on a given day and with a probability of 0.005 if
conditions are normal during the day. Days with dangerous conditions occur with a
probability of y 0.005. Given the above:
(a) What is the probability of a false alarm? I.e. the probability that conditions are normal
when the alarm system triggers?
(b) What is the probability of an unidentified critical condition, i.e. the probability that
conditions are dangerous when the system does not trigger?
Question 13:
Let A and B be two events such that
(i) 𝑃 (𝐴) = 0.4 and
(j) 𝑃 (𝐴 ∪ 𝐵) = 0.7
(a) What is 𝑃 (𝐵) if 𝐴 and 𝐵 are mutually exclusive?
(b) What is 𝑃 (𝐵) if 𝐴 and 𝐵 are independent?
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