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The True Odds Of A Coin Flip

Half dollar coins, like the one pictured above, haven't been minted for circulation since 2002.  Photo by frankieleon (Flickr)

With the odds being 1/2, it would be easy to expect that in a hundred tosses of a coin, you would expect to get about fifty heads.

One Hundred Flips

Ever tried it? Imagine: a kitchen table, a chance idle moment. Flip a penny. Heads.

Toss again. Heads again? Somewhat unusual. Two heads is only one of four possible results in two throws.

Heads again? There's only a one in eight chance of three heads in three tosses.

Heads a fourth time! You've beaten one-in-sixteen odds! Interesting. Keep flipping. Good luck.

You're One In A Million

Twenty heads in a row! No trick coins? The chance of twenty straight heads is about one in a million. Success breeds success, right?

Wrong, in the case of honest coin flips. No matter what happened before, your chance for heads next time is always fifty-fifty.

Ninety-nine tosses, ninety-nine heads! The chance of that is about one in six hundred billion billion billion. If you'd only known, you could be rich!

Honest Coin Flips

At this point, some people might say you're on a roll. Others, that you're due for a string of tails. Smart ones won't play. The chance of getting heads next time is still fifty-fifty.

Two hundred and ninety-nine heads in a row? The odds against that are about one with ninety zeros. Someone should see this: a TV reporter, a mathematician! Too late. The previous two hundred and ninety-eight throws are in the past, gone.

We're out of time. Happy tossing. But remember: honest coin flips are independent. As they say about the stock market: Past performance has no bearing on future results.

Related to this: there are things like buttered toast falling off a table that feels like it should have fifty-fifty odds. Your toast will fall either butter or not-buttered side up. But a physicist demonstrated that with toast, the odds are different than with coins.

Writer's Note:

The way to prove this is to do the math behind the odds. So, The number of possible outcomes of n tosses of a coin is 2n. Straight heads is only one of the 2n possibilities. 220=1.049 x 106; 299=6.338 x 1029; 2299=1.109 x 1090.

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