I think that I will be not the only one and definitely not the first one to claim that recursion is a an educational milestone in Computer Science disciplines. When it comes to understanding recursion, that's one of the "truth moments" in your Computer Science career. If you don't feel comfortable with recursion and don't understand it - you should probably leave the field. Some people claim the same about pointers, but well, even when I find it to be true, nowadays pointer arithmetic plays much less significant role then before, when computer had several hundreds of kilobytes of memory and being able to do pointer arithmetic was considered  a "must master" for any programmer. I guess at a time there was really no other choice at all.

This leads to much more interesting and broader subjects. I remember those times when we were talking of buying 512MB of RAM, creating a RAM-drive and loading there Windows98, so it will run smoother. Well, we were just kids at a time. Nevertheless what I wanted to say was about recursion.

Every CS student has heard at least once during his career "Recursion is cool, however - DON'T use it!". Some might wonder well, why is it so? Well - in fact the recursion is cool indeed, BUT is slow as hell! I have compared just for fun some code that was using a recursion versus non recursive code, and it appeared that non recursive code was about 10 times (!!!!) faster for significant amount of computation. And when I say significant, I mean several hours of runtime.

One must ask - but why? And the answer would be - this is due to stack frames allocations of course. It turns out that it is quite costly to open a new stack frame. I believe there are other reasons as well, which maybe depend on the programming language, but this would be the most obvious one. So kids, don't say I didn't warn you - don't try this at home!

Problem:

How can we choose one out of 3 christmass presents with equal probabability?  Or in other words how to obtain 1/3 probability by using one unbiased coin ?

Solution:

1. The coin is UNbiased: toss the coin twice. Let TH, HT and HH correspond to each of the presents. In case of HH - repeat from the beginning.

2. The coin is biased, then we notice that TH and HT would occur with equal probability but this will only give us 2 possible outcomes with equal probability and we need 3.  Then we would have to use 4 tosses which will give us 16 possible combinations. The question is which ones we should choose to represent the presents (or outcomes). The only condition is that the number of H and T should be the same to ensure equal probability for each outcome and thus selecting each present with the same probability. One option would be to assign HTHT, THTH and HTTH to the three choices, with other  13 possible 4-toss outcomes being rejected.

Discussion:

So the obvious problem is that we need some way to extrapolate tosses of a single coin to (possibly) arbitrary number of outcomes with equal probability to obtain each outcome.  So obviously we will need several tosses to match the number of outcomes we need. However we need to pay careful attention to the fact that the toss sequences we are using have the same combined probability such as HHTT and HTHT

1. How can we choose one out of 3 christmass presents with equal probabability? Or in other words how to obtain 1/3 probability by using one unbiased coin ?
2. Solve the previous problem if the coin is biased and the bias is uknown.

Solution