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You Think a Coin Flip is Always 50-50 Odds? Think Again!

You want a pepperoni and sausage pizza. Your friend wants a veggie-heavy pie. Assuming you’re both too stubborn to go half-meat/half-veggie, flipping a coin ought to be a fair way to decide, right? You would think that a standard coin would have as great a chance at landing on heads as it would on tails, huh?

It turns out that this may not actually be the case, as it depends on context. Let’s say you and your friend are thorough and decide to go best two out of three. Rather, for the purposes of introducing the above video from Numberphile, let’s say that you are waiting for back-to-back flips to come up heads/heads, while your friend is flipping another coin and waiting for back-to-back flips to come up heads/tails.

Let’s start with the basics: The probability of getting consecutive flips coming up heads/heads, heads/tails, or any specific configuration of results is 1 in 4, because there are two possible results for each coin flip, and that’s true over the course of two flips, so 2×2=4.

It turns out that the odds of non-consecutive results coming up (heads/tails or tails/heads) are greater than those of consecutive results (heads/heads and tails/tails). In Dr. James Grime’s experiment, he flipped 50 coins, and found that the expected waiting time for a heads/tails results is four flips, while the expected waiting time for a tails/heads is six flips.

There’s some math involved that’s more complex than that, and Dr. Grime does a fine job at explaining it himself, so watch him do so above, and rest easy knowing that if you can convince your friend to participate in this unusual method of deciding what pizza to get, you have the upper hand.

Image: George Guzenkov

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