2021/8/17 HW3 – Intro to Probability

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Intro to Probability – Homework Assignment 3

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 If 8 identical computers are to be divided among 4 lab rooms, how many divisions are possible? How many if each room must receive at least 1 computer?

Problem 2 Suppose that 10 fish are caught in a pond that contains 5 distinct types of fish.

a. How many different outcomes are possible, where an outcome consists in the number of fish caught of each of the 5 types?

b. How many outcomes are possible if 3 of the 10 fish caught are trout?

c. How many when at least 2 of the 10 are trout?

Problem 3 Prove that

by connecting the formula to a concrete situation (i.e. interpreting its meaning). Show that the formula implies

Problem 4 Consider a tournament of contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players that tied for first place, the next group being those that tied for the next-best position, and so on. Let denote the number of different possible outcomes. For instance, , since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they

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r

n

0

m

r

n

1

m

r − 1

n

r

m

0

( ) = ( ) 2

+ ( ) 2

+ ⋯ + ( ) 2

. 2n

n

n

0

n

1

n

n

n

N (n) N (2) = 3

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could tie for first.

a. List all possible outcomes when .

b. Setting , show that

without resorting to any computation.

c. Find for .

Problem 5 In how many ways can identical balls be placed into urns so that the th urn contains at least balls, for each ? Assume that .

Problem 6 A, B, and C take turns flipping a coin in that order.The first one to get a head wins. Explain why the sample space of this experiment can be defined by

by describing what it means. Then use it to describe the following events:

Problem 7 A retail establishment accepts either the American Express or the VISA credit card. A total of 24% of its customers carry an American Express card, 61% carry a VISA card, and 11% carry both cards. What percentage of its customers carry a credit card that the establishment will accept?

Problem 8 Two cards are randomly selected from an ordinary playing deck (52 cards). What is the probability that one of the cards is an ace and the other one is either a ten, a jack, a queen, or a king? This combinations is called a black jack.

Problem 9 Prove that

n = 3

N (0) = 1

N (n) = n

∑ i=1

( )N (n − i) = n−1

∑ i=0

( )N (i), n

i

n

i

N (n) n = 3, 4

n r i

mi i = 1, … , r n ≥ ∑ r

i=1 mi

1, 01, 001, 0001, 00001, …

WA = ‘‘A wins “, WB = ‘‘B wins “, and (WA ∪ WB) c.

P (E ∩ F c ) = P (E) − P (E ∩ F ).

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2021/8/17 HW3 – Intro to Probability

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Problem 10 Consider an experiment whose sample space consists of a countably infinite number of points. Show that not all points can be equally likely. Can all points have a positive probability of occurring?

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