Does knowing the result of $20$ coin flips change the probability that the coin is biased?

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?

• 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? Commented Sep 15, 2020 at 22:37
• @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? Commented Sep 15, 2020 at 22:48
• 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. Commented Sep 15, 2020 at 22:55
• 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). Commented Sep 15, 2020 at 23:41
• Ah, I see. I'm convinced now, thanks for the explanations Commented Sep 15, 2020 at 23:44