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(solution) Hello! I have a few quick questions on probability, please,


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

 

( 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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