The race between the Democrat presidential hopefuls was so tight in the Iowa caucus Monday that in at least six precincts, the decision on awarding a county delegate came down to a coin toss.
And Clinton won all six, media reports said.
Ok so grab a coin and see if you can come up with the same results six times in a row.
A very impressive achievement indeed.
Especially when you know the odds of such a thing happening.
See how much luck you have.
The odds that Clinton supporters would win all six of the coin tosses against Bernie Sanders supporters are pretty slim.
A lot of people are claiming this is really against the odds, since there’s only a one-in-64 (or a 1.56%) chance of all six coin flips going Clinton’s way. But is that really unusual? Consider the following outcomes, and which ones you would (or wouldn’t) consider unusual:
- Six coin flips: Clinton wins 6, Sanders wins 0.
- Six coin flips: Clinton wins 5, Sanders wins 1.
- Six coin flips: Clinton wins 4, Sanders wins 2.
- Six coin flips: Clinton wins 3, Sanders wins 3.
- Six coin flips: Clinton wins 2, Sanders wins 4.
- Six coin flips: Clinton wins 1, Sanders wins 5.
- Six coin flips: Clinton wins 0, Sanders wins 6.
But 5:1 or 6:0 seems too unlikely to occur at random, doesn’t it?
Unfortunately, this is one of those cases where our mathematical intuition and what actual probabilities are don’t line up at all.
If you have a fair (50/50) coin at play in each instance, it’s true you’re more likely to have three “wins” for each candidate than any other specific outcome.
But it’s still not all that likely: there’s only a 31.25% chance that Clinton and Sanders would have walked away with three delegates apiece.
Furthermore, the odds that Clinton would win four and Sanders would win two is only a little worse: 23.44%.
But if you combine that with the odds that Sanders would’ve won four with Clinton winning two, you get that a 4:2 outcome has a 46.88% chance of happening.
Meaning the “unlikely” outcomes of 5:1 or 6:0? They actually have a 21.88% chance of occurring, which is about the same as your odds of winning any prize at all (most likely, $4) if you buy six random Powerball tickets.
Ok so the cynics among us believe there is a lot of cash being passed around behind the scenes...LOL!
And those magic, weighted coins are in evidence?
Diaconis is a professor of mathematics
and statistics at Stanford University and, formerly, a professional
magician.
While his claim to fame is determining how many times a deck of cards must be shuffled in order to give a mathematically random result (it’s either five or seven, depending on your criteria), he’s also dabbled in the world of coin games.
What he and his fellow researchers discovered (here’s a PDF of their paper) is that most games of chance involving coins aren’t as even as you’d think.
For example, even the 50/50 coin toss really isn’t 50/50 — it’s closer to 51/49, biased toward whatever side was up when the coin was thrown into the air.
But more incredibly, as reported by Science News, spinning a penny, in this case one with the Lincoln Memorial on the back, gives even more pronounced odds — the penny will land tails side up roughly 80 percent of the time.
The reason: the side with Lincoln’s head on it is a bit heavier than the flip side, causing the coin’s center of mass to lie slightly toward heads.
The spinning coin tends to fall toward the heavier side more often, leading to a pronounced number of extra “tails” results when it finally comes to rest.
Because the coins typically pick up dirt and oils over time, trying the experiment at home may not yield such a large percentage of “tails” over “heads” — but a relatively new coin should still give you noticeable results.
While his claim to fame is determining how many times a deck of cards must be shuffled in order to give a mathematically random result (it’s either five or seven, depending on your criteria), he’s also dabbled in the world of coin games.
What he and his fellow researchers discovered (here’s a PDF of their paper) is that most games of chance involving coins aren’t as even as you’d think.
For example, even the 50/50 coin toss really isn’t 50/50 — it’s closer to 51/49, biased toward whatever side was up when the coin was thrown into the air.
But more incredibly, as reported by Science News, spinning a penny, in this case one with the Lincoln Memorial on the back, gives even more pronounced odds — the penny will land tails side up roughly 80 percent of the time.
The reason: the side with Lincoln’s head on it is a bit heavier than the flip side, causing the coin’s center of mass to lie slightly toward heads.
The spinning coin tends to fall toward the heavier side more often, leading to a pronounced number of extra “tails” results when it finally comes to rest.
Because the coins typically pick up dirt and oils over time, trying the experiment at home may not yield such a large percentage of “tails” over “heads” — but a relatively new coin should still give you noticeable results.
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