Probability shortlist
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? (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)
min = p[0]
for x in p:
if min > x:
min = x
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