Life's been keeping me busy. I'll post an update soon. Briefly: finished my summer job for the Amalgamated Insight (née TelegraphCQ) folks. I'm now back at Queen's for the last year of my undergrad.
Interesting math
An assignment in one of my classes included an interesting bonus problem. It is very simple, but I confess I got it completely wrong before I saw the solution. Maybe one of you bright folks is smarter than I:
Let the alphabet A = {a, b, c, ..., z} (A is the set of 26 lowercase letters of the English alphabet). Let S1(w) be true iff the string w over alphabet A contains the substring aaa; let S2(w) be true iff the string w contains the substring abc.
Suppose we choose a w of 10 characters; each character in w is selected randomly and independently.
Let P1 be the probability that S1(w) is true, and let P2 be the probability that S2(w) is true. Is P1 > P2, P1 < P2, or P1 = P2? Give a justification for your answer. (Hint: P1 != P2).
If you think you know the answer, email me — I'll post the answer later (neilc A T samurai D O T com). Obviously, the gist is in the justification, not which alternative you think is true.
Hat Tip: Prof. Kai Salomaa for showing me the problem.