Solved – How to 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?

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 $p=19/22simeq0.86$ 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 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.

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