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Hello! I have a few quick questions on probability, please,**More:**

**Hello! I have a few quick questions on probability, please, numbers 1, 3, 6, 7, 8. I figured out the ones in the book. Thank you!**

Math 180A - Homework 2

Math 180A - Homework 2

( Due Friday, OCT 7, 2:00 PM)

Reading: Sections 1.1 - 1.5 and 2.5 of the textbook and lecture notes 4-6.

Show FULL JUSTIFICATION for all your answers. 1. In how many ways can 8 people be seated in a row if

a. there are no restrictions on the seating arrangement;

b. persons A and B do NOT want to sit next to each other;

c. there are 4 married couples and each couple must sit together.

2. Problem 17. of Chapter 2. of the textbook.

3. An urn contains 12 red balls and 6 black balls. If we choose 5 balls at random from the urn,

what is the probability that at least 2 black balls will be among the 5 balls chosen?

4. Problem 31. of Chapter 2. of the textbook.

5. Problem 29. of Chapter 2. of the textbook.

6. Provide a purely combinatorial argument (requiring little or no computations) to establish

the following identity for an integers n ? 1:

X

n

i?1

n

=

k?1

k

i=k (Hint: consider the event Ai that when you choose k numbers from the set {1, ..., n}, the

largest number chosen is i)

7. Provide a purely combinatorial explanation of why

n

n

=

r

r, n ? r

8. a. Suppose 8 cards are chosen at random from one deck of playing cards. What is the

probability that exactly 2 cards are chosen from each suit.

b. Now suppose we have 8 sets of playing cards, and we choose one card at random from

each set. What is the probability that, among 8 cards chosen, there are exactly 2 cards

from each suit.

c. Compare the results of parts a and b: which of the two cases has a bigger probability?

Provide a heuristic argument for why such is the case. 9. Problem 54. of Chapter 2. of the textbook.

10. Fill out the Survey No. 2 in TritonEd. 1

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