How do I use math to predict the next number in the series? Here's a series of data I'm observing:
1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1

How do I use math to predict whether the next number in the series will be a 1 or a 0?
 A: If observations are independent, and if values must either be 1 or 0, with no additional prior information, you may simply assume that the probability that the next value is 1 is equal to the proportion of 1s in the observations.
If you wish to calculate a confidence interval around this estimate, this could reasonably be modeled as a Bernoulli trial with probability $p=19/22\simeq0.86$ And a  95% confidence interval of $[65\%,97\%]$ (CI calculated as the Clopper-Pearson interval). 
This model is analogous to expecting heads from a coin that has landed on heads in 19 of 22 flips, or drawing a white pebble from a bag where the previous 22 draws gave 19 white + 3 nonwhite pebbles (if the pebbles are put back each time, or if there are infinite well mixed pebbles).
See also https://stats.stackexchange.com/a/6184/1381 for information and alternative methods for computing confidence intervals for Bernoulli trials. 
Given the number of up votes on the OP, perhaps there is a less trivial solution, but I suspect that it just looks like it would be interesting if the observations were related, and order mattered, rather than being independent.
