How to know which coin land on head the most given a data set? Suppose I have 2 coins that I don't know what the probability of it landing on head is. It can be even (i.e. 50%) or it could be 2 biased coins that lands on heads most of the time.
I do not have access to those coins but someone did some flipping for me.
The person got lazy and only did 200 flips for first coin and 80 flips for the second coin.
The result of the first coin after 200 flips:
141 heads, 59 tails
The result of second coin after 80 flips:
71 heads, 9 tails

*

*How do I go about finding the true probability of landing on heads for each coin?


*Suppose the true probability of the second coin landing on head is higher than the first, how do I take the extra trials in the first coin into account when deciding which coin would land on heads the most?
 A: *

*You don't. The true probability of heads is unknowable, and can only be estimated from observed data. This estimate can have an arbitrarily narrow confidence interval given an arbitrarily large number of flips, but you can never know the exact probability of getting heads for either coin just from observed data.


*The best estimate of the probability of heads for each coin is simply the observed proportion of heads for each coin. No matter how many times you've flipped the coins, the one that comes up heads more frequently is the one with the higher probability of getting heads. Flipping more times narrows the confidence intervals around the predicted probabilities, meaning you can be more sure that one coin has a higher probability and not the other, but the observed ranking is your best guess no matter how many times you've flipped. If heads comes up 50% of the time for Coin A and 60% of the time for Coin B, it's likely that Coin B has the higher probability of heads. That's true whether you flipped the coin 10 times or 10,000 times (or even if you flip one 10 times and the other 10,000 times), although you'll be more sure of your conclusion the more flips you do.
A: You can never know the true distribution after a finite number of trials. However, you can calculate a binomial proportion confidence interval for your experiment.
I like to use Wilson's method for this. It gives a 95% confidence interval of $[0.638, 0.764]$ for the first coin and $[0.800, 0.940]$ for the second.
