Ah, the Bernie Sanders camp is crying Foul this morning:

Then there is the fact that in at least six precincts (that we know of so far) Hillary attained a bare majority of votes literally on a coin toss. Which is to say that the Clinton side called the coin flip accurately six times, the odds of which are

64:1 against. Which is to say a 1.5 percent chance.

However, that calculation would seem valid only if the same person (or candidate's rep) made the call all six times. But Hillary-camp did not make all six calls. The WaPo mentions one toss that was called (wrong)

by Sanders' side and presumably there were others.

If each side got to call three tosses, then what are the odds that Clinton would still win all six? Another way of asking is, What are the odds that Sanders' side would choose wrong three times and Clinton's choose right three times?

For Sanders' side to choose wrong the first time is one chance in two, or .5. The same chance pertains for the other two flips, also, so the odds stack thus in getting all three flips wrong: .5 X .5 X.5, which equals .125, or 12.5 percent (8:1). That does not seem to dauntingly out of the realm of possibility. In fact, if the Powerball lottery had winning odds of 8:1, I'd buy nine tickets every time! Of course, the same calculation is used for Hillary's odds of choosing the winning toss three times.

So there is a one in eight chance that Sanders will choose his three flips wrong, and the same that Hillary will choose her three flips right. Combined, the chance of Hillary winning all six tosses are .125 X .125=.015, right? So doesn't that put us back to where we started?

Nope. Each toss is an independent event: one toss coming up heads neither increases nor decreases that chance that a subsequent toss will land either way, nor does it affect the probability that the next call will be more or less likely to be right or wrong. There were six separate tosses, each with a .5 chance of landing as called.

The second reason is that while it might make sense to link each candidate's prediction together to multiply probabilities, I do not think you can cross from Sanders to Hillary and link their respective probabilities together. There were, in a sense, not six coin tosses that Hillary won, there were two sets of three tosses. Hillary won one set and Sanders lost the other, and those two outcomes are not necessarily probabilistically related, even though they are politically related.

Further complicating this is the fact that the actual calculation needed is

*not* the odds of the coins falling heads or tails. The coins landed sometimes one, sometimes the other. The odds that need to be calculated are not those of heads or tails, but the odds that the caller will

*accurately predict which*.

And this teaches us why intuition is not reliable when calculating probability. There is a 50-50 chance that a toss will fall heads. But there is also a 50-50 chance that a caller will call heads in advance. So intuitively, it would seem that you have right off the bat a .5 X .5 = .25, or 25 percent chance that the call will be accurate. And then for three calls in a row, .25*.25*.25, or .015 chance that all three calls will be right, meaning that Hillary's win of all tosses is a truly astounding .015*.015, or 0.000225 (0.0225 percent)!

Except it isn't. You can see why easily by charting the pairing of possible calls and with possible outcomes:

You can see that the chance remains 0.5 to make the correct call. There are four possible call/fall pairings and the caller wins two and loses two.

*Update*: Is it an error to connect these coin tosses as a sequence? I could argue that they are not. They were in fact independent events that each had nothing to do with each other - separate coins, separate players, separate locations, different times. Hillary's team called some and Sanders' team called others. Hillary in fact did not win six tosses; she won some and Sanders lost the others, as I explained above. There's a difference.

There may be no more reason to chain these together as a mathematically-linked sequence than to link her correct calls with the odds that a NY cabbie could make a run through three or four consecutive green lights.

But someone out there with actual training in probability and statistics, leave a comment!

*Update*: Now news media are reporting that there were many more coin tosses than just these six.

The WaPo:

The initial 6-for-6 report, from the Des Moines Register missed a few Sanders coin-toss wins. (There were a lot of coin tosses!) The ratio of Clinton to Sanders wins was closer to 50-50, which is what we'd expect.

And remember: