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Our friend has a $50\%$ chance of giving us a regular coin and a $50\%$ chance of giving us a biased coin that flips heads $75\%$ of the time.

We take the coin, and the probability that it's biased is obviously $50\%.$ But now, we flip it $20$ times and get $18$ heads. What is the probability that the coin is biased?

I think that the probability that the coin is biased actually doesn't depend on what we see after getting the coin, but I could also use Bayes' Theorem to calculate $P(\text{biased} \mid \text{$18$ heads out of $20$ flips}),$ so which one is right?

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    $\begingroup$ Why do you think that observing lots of outcomes from the coin won't help you distinguish between the two possibilities? If your friend gave you a toaster and said that one slot burns the toast 50% of the time and the other slot burns it 75% of the time (but they couldn't remember which slot was which), and you found that the left slot burned the toast 18 of the first 20 times, wouldn't you feel pretty good that it was the right slot that only burns toast 50% of the time? $\endgroup$
    – Greg Martin
    Commented Sep 15, 2020 at 22:37
  • $\begingroup$ @GregMartin yeah it makes sense that the coin is probably biased... but how can the probability of something that happened before we tested the coin change afterwards? What am I misunderstanding that makes me feel like this is rewriting history? $\endgroup$
    – ocw05
    Commented Sep 15, 2020 at 22:48
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    $\begingroup$ It’s not rewriting history, you’re just learning more about the past in the present. Bayesian probability is subjective; it’s about what you can conclude based on what you’ve seen. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 15, 2020 at 22:55
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    $\begingroup$ Or, to put it another way: the probability that you received the biased coin is either 100% (if you received it) or 0% (if you didn't receive it). Even before making any tosses, the "50% probability" that you received the biased coin is a statement about your perceptions and conclusions, not about actual history. So updating your perception-conclusion probability isn't rewriting history either. It's legitimately confusing that "probability" can be used both in this personal-conclusions way and also to talk about statistical predictions of the future (like a 75%-heads coin). $\endgroup$
    – Greg Martin
    Commented Sep 15, 2020 at 23:41
  • $\begingroup$ Ah, I see. I'm convinced now, thanks for the explanations $\endgroup$
    – ocw05
    Commented Sep 15, 2020 at 23:44

1 Answer 1

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Perhaps you are confusing two probabilities.

First there is the probability that a particular coin when flipped shows heads. This is 50% for the fair coin and 75% for the biased coin. This probability does not change unless something about the coin is changed. So, for example, even after flipping the fair coin 100 times and getting 100 heads, the probability that the next flip will be heads is still 50%.

Second there is the probability that the coin you have been given is the biased coin. This is a measure of your confidence about which coin you have been given. If you get more information your confidence about which one it is may change. For example, if you learn that the fair coin is silver and the biased coin is bronze then you can become 100% confident about which coin you have just by looking at it. The coin doesn't change - it is your confidence about which coin you've been given which changes.

Initially you have no reason to believe that the coin is fair or biased. All you know is that it must be one or the other, the coins look identical, and it is equally likely to be either. So you assess the probability that it is biased to be 50%. This is clearly different to the probability that the coin will show heads when flipped.

Now you flip the coin 20 times and get 18 heads. You have gained some information about the coin in your possession. This increases your confidence that it is the biased coin because this outcome is more consistent with it being the biased coin. You might now be 90% confident that it is the biased coin. But you cannot be 100% sure because this outcome is also possible - but highly unlikely - if the coin is fair.

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