A run or a streak is a series of the same outcome in a random sample. For example 3 heads in a row in a series of 10 coin flips, or 1000 votes in a row for a single candidate.

Calculating the exact answer requires going over the canonical ensemble and is not practical for large samples, but a proof is done here. However there is a good approximation, as follows:

N is the number of trials, K is the minimum streak size, p is the probability of a single event, and q is the complement, 1-p.

Assume that N is pretty big compared to K. A string of heads (that can be zero heads long) starts with a tails, and there should be about Nq of those. The probability of a particular string of heads being at least K long is p^K so you can expect that there should be around E=Nqp^K strings of heads at least K long. When E≥1, that means that it’s pretty likely that there’s at least one run of K heads. When E<1, E=Nqp^K is approximately equal to the chance of a run of at least K showing up.

Lets put some numbers to this and see what happens

For N=1000, p=q=.5, and K =10

1000*.5*.5^10 = 0.488 or about a 50% chance of getting a string of 10 heads in 1000 coin flips

For N=10,000, p=q=.5, and K =100

10,000*.5*.5^100 = 3.94*10^-27

Already we can see that long streaks are very uncommon . You would be just as likely to win the Powerball lottery three draws in a row ((10^-9)^3 or 10^-27).

Some time tonight, we will be asked to believe that a streak of 1000 showed up in 100,000 votes

For N=100,000, p=q=.5, and K =1000

100000*.5*.5^1000 = 4.67*10^-297

and for those of you that say votes are not coin flips

For N=100,000, p=.6, q=.4, and K =1000 giving a 10% advantage to Mcluffie, completely unrealistic.

100000*.4*.6^1000 = 5.67*10^-218

This is at the same level as winning every single Powerball lottery for an entire year. It is an event that will never occur in a universe of universes. So either they either are ruling by divine right, or they are stealing our government.