Probability shortlist

Cosmin Negruseri
18 iunie 2014

Here's a set of probability problems. Try to solve them in the comments section.

• (von Neumann’s biased coin) You’re given a biased coin. It falls heads with an unknown probability p and tails with the probability 1 - p. Can you come up with a way to generate fair 0 and 1 results?
• A rand5() function gives uniform integer results between 1 and 5. How can you use it to build a rand7() function?
• Build a function rand(x, y, r) that returns a uniformly random point in the circle given by the (x,y) center and the r radius.
• What’s the expected number of times line x is executed if p is a uniform random permutation of numbers 1 .. n.

min = p[0]
 for x in p:
   if min > x:
     min = x

• Build an algorithm that returns uniform random permutations.
• Given a stream of integers build an algorithm that returns a uniform random sample of size k using O(k) memory.
• (Coupon collector’s problem) Suppose a kid wants to collect all the cartoon characters in a kinder surprise series. Given that there are n different characters in total and they are distributed uniformly random
• (Balls and bins problem) m balls are thrown into n bins. Each ball has 1/n probability of falling into each bin. What’s the expected number of empty bins?