As most of you probably know, I'm very much into odds and handicapping and the like. I got into an argument with a random stranger about the Pick 3 lottery on the bus, and it was very maddening because I found that while I am absolutely convinved that he is wrong and I am right, I cannot actually demonstrate this because I lack the probability background.
I won't describe the argument itself for fear of prejudicing you, but here's the question:
If I run the "Pick 3" experiment over and over again, eventually I will observe every possible outcome from 000 to 999. The question is, how many times, on average, should I expect to have to run the experiment before I have observed all those results? Or, to put it a different way, what is the percentage chance that I will achieve all 1000 results before I have run the experiment 10,000 times?
I can actually come up with a pretty close approximation of the answer to this question by writing a program that defines an array with 1000 entries in it with all of the entries set to 1 initially, and whenever the test returns a number, the corresponding spot in the array is set to zero. The program terminates and reports on its results once the sum of all the numbers in the array is zero.
So I'm actually less interested in the actual answer to the question as the "how" of answering the question. I need to know the formula that an actual probablity person would use to determine this algebraically.
At issue is the correctness of my guiding maxim as a gambler, that no betting strategy can give a series of negative expectation bets an overall positive expectation.