Since lately my post on the twin coin flipping “paradox” has been gaining attraction, and there are still misconceptions about the outcome, I thought I should hack up a quick script to demonstrate. Experimenting is always the best way to verify an idea so let’s do that.

First off, a quick review; the problem is: flip two coins simultaneously. If one coin is heads, what is the probability that the other coin is also heads?

Notice how the statement does not say that the FIRST or the SECOND coin is heads. It merely says that one of them is heads. And therein lies the difference in the outcomes. If we know the first coin is heads, then the probability that the other is also heads is in fact 0.5 (or 50%). If we know that (at least) one of them is heads (not any one specifically), then the probability that the other is also heads is 0.33 (or 33%). This might sounds counter-intuitive, but lets brake it down and see.

There are four possible combinations for this outcome: H-H, H-T, T-H, and T-T. If we know one is heads, then the only possibility that is eliminated is T-T. Hence, the probability that the other one is also heads, its 1/3 since only one of the remaining combinations satisfy our requirement.

To demonstrate, I have hacked a small javascript. Set the number of two-coin flippings you want to perform, and the script will randomly assign heads or tails to each coin, and calculate the frequencies of appearance for each of two scenarios: i) one coin is heads, ii) the first coin is heads. You can find the simple source here.

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