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 headsin a row does indeed have much less than 0.5 chance, to be precise 0.521, 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 0.520. 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 0.521, 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 0.5.
So the thing is something like the following, right?: “Looking at it from the outside, a world where 21 heads showed in a row is incredibly unlikely: (if the coin is fair) I would happily bet against this world happening. However, I am already in an incredibly weird world where 20 heads have shown in a row, and another heads only makes it a bit more weird, so I don’t know what to bet, heads or tails.”
Yes, essentially. While 21 heads in a row is very unlikely (when you consider it ahead of flipping any coins), by the time you get to 20 heads in a row most of the unlikely-ness of it has already happened, with the odds of one more head remaining the same as ever.
The advice is: do not bet. Suppose you download a gambling app that bets on games where the outcome is similar to a coin flip. You start receiving emails from someone associated with the app (so they bypass your spam filters). Each day for 20 days you receive an email predicting the outcome of the game. Each of the 20 predictions is correct. What do you do? Nothing. What you are unaware of (but should suspect) is that on the first email, the sender has sent out 8 million emails making a prediction (it is a popular gambling app). 4 million of those predicted the home team wins and the other 4 million predicted the visiting team wins. The next day the emails only goes out to those that received the correct prediction. Rinse. Repeat. And you happen to be an (un)lucky recipient of the 21st email distribution. The world you live in is no weirder than the world a Powerball Lottery winner lives in.
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 0.521, 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 0.520. 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 0.521, 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 0.5.
P(21 heads)=0.521=0.000000476837158
P(21 heads|20 heads)=0.5
Let me know if any of that is still confusing.
I think you explain it very well!
So the thing is something like the following, right?: “Looking at it from the outside, a world where 21 heads showed in a row is incredibly unlikely: (if the coin is fair) I would happily bet against this world happening. However, I am already in an incredibly weird world where 20 heads have shown in a row, and another heads only makes it a bit more weird, so I don’t know what to bet, heads or tails.”
Yes, essentially. While 21 heads in a row is very unlikely (when you consider it ahead of flipping any coins), by the time you get to 20 heads in a row most of the unlikely-ness of it has already happened, with the odds of one more head remaining the same as ever.
The advice is: do not bet. Suppose you download a gambling app that bets on games where the outcome is similar to a coin flip. You start receiving emails from someone associated with the app (so they bypass your spam filters). Each day for 20 days you receive an email predicting the outcome of the game. Each of the 20 predictions is correct. What do you do? Nothing. What you are unaware of (but should suspect) is that on the first email, the sender has sent out 8 million emails making a prediction (it is a popular gambling app). 4 million of those predicted the home team wins and the other 4 million predicted the visiting team wins. The next day the emails only goes out to those that received the correct prediction. Rinse. Repeat. And you happen to be an (un)lucky recipient of the 21st email distribution. The world you live in is no weirder than the world a Powerball Lottery winner lives in.
That’s a nice example. I heard about it long ago with investments instead of games. It is really something important to keep in mind!
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)).)