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.

# (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? (microsoft interview)
# 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. (dropbox interview)
# Build an algorithm that returns a uniform random permutation of numbers 1 to n. (google interview)
# (Reservoir sampling) Given a stream of integers build an algorithm that returns a uniform random sample of size k using O(k) memory. (google interview)
# (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. What's the average number of kinder eggs he has to buy so that the selection is complete. (twitter interview)
# (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? (twitter interview)
# What’s the average number of times line 4 is executed if p is a random permutation of numbers 1 to n. (pinterest interview)
 

==code(python) |
 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?

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