If you’re interested in making a follow-up post, I’d enjoy an analysis of the possibilities when the coin is not fair but is also not double sided. For example, if a coin has a 75% chance of turning up heads, how does the probability look? If a coin turns up heads 50 times in a row, it’s probably neither fair nor a 75⁄25 coin, but if it turns up heads 10 times in a row I might guess it to be 75⁄25.
If you’re interested in making a follow-up post, I’d enjoy an analysis of the possibilities when the coin is not fair but is also not double sided. For example, if a coin has a 75% chance of turning up heads, how does the probability look?
I wrote this! The graphs of P(bias|flips) are fun. See this post starting at “computing a credible interval”:
A string of all-heads makes “the coin always flips heads” more likely than any other option, given equal priors, no matter how long the string is.
So, what is your prior distribution of bias for “a coin someone tells you to flip”?
I’d say 1000:10:1:.001 for fair:biased a tiny but detectable amount:always heads:any other bias amount
I’ve read that it’s not possible to bias a coin—you can bias a coin toss if you know which way up it starts, but the coin itself will always be fair. But I confess that I don’t know what assumptions they were making, so for all I know you could make something that would be recognizably a coin but that analysis wouldn’t apply.
If one side is heavier, it will land that side down more often. You can see this with a household experiment of gluing a quarter to a circle of cardboard the same thickness, and then flipping it.
So I was thinking of this paper (pdf), which I misremembered somewhat—you can’t make a coin biased for “toss and catch”, but you can make it biased for “toss and let it bounce”. (And for “spin on a table”.) Given that, “can’t bias a coin” is probably too strong, though it’s in the title of the paper.
Props for suggesting an actual experiment! I didn’t feel like doing it though :p
If you’re interested in making a follow-up post, I’d enjoy an analysis of the possibilities when the coin is not fair but is also not double sided. For example, if a coin has a 75% chance of turning up heads, how does the probability look? If a coin turns up heads 50 times in a row, it’s probably neither fair nor a 75⁄25 coin, but if it turns up heads 10 times in a row I might guess it to be 75⁄25.
I wrote this! The graphs of
P(bias|flips)
are fun. See this post starting at “computing a credible interval”:https://justinpombrio.net/2021/02/19/confidence-intervals.html
Sorry if you’re viewing on mobile, I need to fix my styling.
A string of all-heads makes “the coin always flips heads” more likely than any other option, given equal priors, no matter how long the string is. So, what is your prior distribution of bias for “a coin someone tells you to flip”? I’d say 1000:10:1:.001 for fair:biased a tiny but detectable amount:always heads:any other bias amount
I’ve read that it’s not possible to bias a coin—you can bias a coin toss if you know which way up it starts, but the coin itself will always be fair. But I confess that I don’t know what assumptions they were making, so for all I know you could make something that would be recognizably a coin but that analysis wouldn’t apply.
If one side is heavier, it will land that side down more often. You can see this with a household experiment of gluing a quarter to a circle of cardboard the same thickness, and then flipping it.
So I was thinking of this paper (pdf), which I misremembered somewhat—you can’t make a coin biased for “toss and catch”, but you can make it biased for “toss and let it bounce”. (And for “spin on a table”.) Given that, “can’t bias a coin” is probably too strong, though it’s in the title of the paper.
Props for suggesting an actual experiment! I didn’t feel like doing it though :p