## Probability One

 Saying something happens with probability 1 does not mean it always happens; saying something happens with probability 0 does not mean it never happens Rather, suppose the process is repeated again and again. We compute the fraction If the fraction goes to 1 as the total number of repetitions goes to infinity, the event happens with probability 1. If the fraction goes to 0 as the total number of repetitions goes to infinity, the event happens with probability 0. Under forced circumstances, this perfectly sensible definition can lead to some counterintuitive results. For example, suppose a coin is tossed infinitely many times, and heads comes up on only the 1st, 2nd, 4th, 8th, 16th, 32nd, 64th, 128th, 256th, 512th, 1024th, ... tosses. That is, heads comes up only on the tosses numbered by powers of 2, and all other tosses give tails. Here we plot the fraction (vertically, the top of the vertical line is at 1) vs n for n to 1024. The fraction goes to 0, so we are left with the odd situation that heads comes up infinitely many times, yet the probability of getting heads is 0. So: probability = 0 does not mean an event never occurs, just that it is increasingly unlikely with more repetitions. Also, probability = 1 does not mean an event always occurs, just that is is increasingly likely with more repetitions.