I received a lot of comments on my post on the probability of Hillary Clinton winning all six coin tosses in a tied caucus site is 1 in 64 or 1.56% or 0.0156 as the Des Moines Register reported. NPR cited an unnamed Democratic party official saying that there were a dozen coin tosses and Sanders won "at least a handful" while other media outlets and the Sanders campaign have repeated the claim that there were six. Someone sent me the above video which appears to show Sanders winning a coin toss.
Lets be conservative and say that there were seven coin tosses with Sanders winning one. What is the probability of this outcome? According to the binomial distribution, the probability of exactly one success out of seven coin tosses (with the chance of success on 1 toss being 0.5) is 0.0547 or 1 in 18.28. The probability of 1 or fewer successes is 0.0625 or 1/16.
If we take the democratic official at his word that he won "a handful" out of a dozen tosses (we will call a handful 5), the probability of exactly 5 successes in 12 tosses is 0.1934 or about 1 in 5. The probability of 5 or fewer successes in 12 tosses is 0.381 or 38.7% or 1 in 2.58 tosses.
The probability in the first two scenarios is low but there is a greater chance of these scenarios than dying in a plane crash (1 in 5.4 million), being struck by lightning in one's lifetime (1 in 12,000), or the probability of winning the Powerball jackpot (1 in 175,223,510). The Des Moines Register still says that there were irreguarities in the caucus process and that the gap between Clinton and Sanders narrowing to 700.47 state delegates for Clinton and 696.92 for Sanders.
Hopefully New Hampshire;s primary won't have the same issues. Other states do have caucuses similar to Iowa. Voting or caucusing irregularities are usually noticed when the results are as close as Iowa this year or Florida in 2000. Results that are independently verifiable is the key.