概率论 (Summer 2013)/Problem Set 1

Problem 1

(Due to J. von Neumann.)

1. Suppose you are given a coin for which the probability of HEADS, say [math]\displaystyle{ p }[/math], is unknown. How can you use this coin to generate unbiased (i.e., [math]\displaystyle{ \Pr[\mbox{HEADS}]=\Pr[\mbox{TAILS}]=1/2 }[/math]) coin-flips? Give a scheme for which the expected number of flips of the biased coin for extracting one unbiased coin-flip is no more than [math]\displaystyle{ 1/(p(1-p)) }[/math].
2. Devise an extension of the scheme that extracts the largest possible number of independent, unbiased coin-flips from a given number of flips of the biased coin.

Problem 2

(Due to D.E. Knuth and A. C-C. Yao.)

1. Suppose you are provided with a source of unbiased random bits. Explain how you will use this to generate uniform samples from the set [math]\displaystyle{ S=\{0,\dots,n-1\} }[/math]. Determine the expected number of random bits required by your sampling algorithm.
2. What is the worst-case number of random bits required by your sampling algorithm? Consider the case when [math]\displaystyle{ n }[/math] is a power of [math]\displaystyle{ 2 }[/math], as well as the case when it is not.
3. Solve (1) and (2) when, instead of unbiased random bits, you are required to use as the source of randomness uniform random samples from the set [math]\displaystyle{ \{0,\dots,p-1\} }[/math]; consider the case when [math]\displaystyle{ n }[/math] is a power of [math]\displaystyle{ p }[/math], as well as the case when it is not.

Problem 3

(Due to D.R. Karger and R. Motwani.)

1. Let [math]\displaystyle{ S,T }[/math] be two disjoint subsets of a universe [math]\displaystyle{ U }[/math] such that [math]\displaystyle{ |S|=|T|=n }[/math]. Suppose we select a random set [math]\displaystyle{ R\subseteq U }[/math] by independently sampling each element of [math]\displaystyle{ U }[/math] with probability [math]\displaystyle{ p }[/math]. We say that the random sample [math]\displaystyle{ R }[/math] is good if the following two conditions hold: [math]\displaystyle{ R\cap S=\emptyset }[/math] and [math]\displaystyle{ R\cap T\ne\emptyset }[/math]. Show that for [math]\displaystyle{ p=1/n }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.
2. Suppose now that the random set [math]\displaystyle{ R }[/math] is chosen by sampling the elements of [math]\displaystyle{ U }[/math] with only pairwise independence. Show that for a suitable choice of the value of [math]\displaystyle{ p }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.

Problem 4

(Due to R.M. Karp)

Consider a bin containing [math]\displaystyle{ d }[/math] balls chosen at random (without replacement) from a collection of [math]\displaystyle{ n }[/math] distinct balls. Without being able to see or count the balls in the bin, please devise a strategy to simulate random sampling with replacement from the original set of [math]\displaystyle{ n }[/math] balls. Notice that your only access to the balls is that you can sample without replacement from the bin.

Problem 5

We play the following game:

Start with [math]\displaystyle{ n }[/math] people, each with 2 hands. None of these hands hold each other. At each round, uniformly pick 2 free hands and let these two hands hold together. Repeat this until no free hands left.

• What is the expected number of cycles made by people holding hands with each other (one person with left hand holding right hand is also counted as a cycle) at the end of the game?