Jalen Lyle-Holmes

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Yep that's it! Glad my explanation helped.
(Though if we want to be a bit pedantic about it, we'd say that actually a world where 21 heads in a row ever happens is not unlikely (If heaps and heaps of coin tosses happen across the world over time, like in our world), but a world where any particular given sequence of 21 coin flips is all heads is yes very unlikely (before any of them have been flipped)).)

Ah yes this was confusing to me for a while too, glad to be able to help someone else out with it!

The key thing to realise for me, is that the probability of 21 heads in a row changes as you toss each of those 21 coins.

The sequence of 21 heads in a row does indeed have much less than 0.5 chance, to be precise ,   which is 0.000000476837158.  But it only has such a tiny probability before any of those 21 coins have been tossed. However as soon as the first coin is tossed, the probability of those 21 coins all being heads changes. If first coin is tails, the probability of all 21 coins being heads goes down to 0, if first coin is heads the probability of all 21 coins being heads goes up to . Say you by unlikely luck keep tossing heads. Then with each additional heads in a row you toss, the probability of all 21 coins being heads goes steadily up and up, til by the time you've tossed 20 heads in a row, the probability of all 21 being heads is now.... 0.5, i.e. the same as a the probability of a single coin toss being heads! And our apparent contradition is gone :)

The more 'mathematical' way to express this would be: The unconditional probability of tossing 21 heads in a row is , i.e. 0.000000476837158 but the probability of tossing 21 heads in a row conditional on having already tossed 20 heads in a row is .



Let me know if any of that is still confusing.

 

Love this way at pointing at this distinction!

Oooo thanks for this, just used it to write a post for my blog and it was more fun and easier than usual: My Anxiety Keeps Persisting Even Though It Has Been Diagnosed, Rude 

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