Who Were the Sea Peoples?

One-dimensional drunkard's walk: Starting from Point X, you stumble discretely east (probability 1/2) and west (probability 1/2) and repeat, indefinitely. Successive stumbles are unrelated, i.e. , the former does not in any way predict the latter (nor vice versa ). In the long run, what is the average number of stumbles that will bring you back to Point X.

What is 1/1 + 2/2 + 3/4 + 4/8 + 5/16 + 6/32 + ...?

Thanks to Prof. Burton Dreben (Math 281, 1961): Is there a {computer, computer program, algorithm, etc. } which solves the following problems, i.e. , are these problems decidable in the metamathematical sense? a ) Are there 777 consecutive 7's in the decimal expansion of pi? b ) Is there exactly one God? Or are one or both of these questions perhaps unsolvable?

Welcome to a new, short series of problems, presented here for your, uh, not sure what... Say you have access to a repeatable experiment whose favorable outcome, utterly randomly determined, is very unlikely--like a state lottery. Say the likelihood of "winning" (whatever that means) is 1 in 1,000,000,000. OK, you repeat it 1,000,000,000 (same number as above) times. What is the likelihood that you still haven't won it even once? What is the likelihood that you have won it exactly once. (1 minus the sum of those two is the likelihood that you have won it two or more times, ending with the very, very small probability that you have won it every single time, or 1,000,000,000 times in all.)