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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

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

  2. 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?

jun
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    This is not clear. If you have no a priori concept of the two probabilities, you really can't do better than to take the empirical probabilities, so $\frac {141}{200}$ for the first and $\frac {71}{80}$ for the second. If you are assured that the probability for the second is higher than for the first, then you expect the second to come up $H$ more often in the same number of trials. Not sure what you are asking there. – lulu Feb 22 '22 at 21:23
  • Is there a better way to estimate the true probability of each coin other than the naïve approach of 141/200 and 71/80? What if the results are purely luck and the probability of each coin landing on heads are actually close to 50% or even 55%? – jun Feb 22 '22 at 21:25
  • You can't even know which coin is which by looking at two sets of trials. The "fair" coin could give you something absurd like 70 heads and 10 tails, while the "biased" coin could give you 40 heads and 40 tails. We know that both of these outcomes would be unlikely, but we can't know for certain which coin is fair and which is biased based on trials alone. We could make an educated guess about which coin is fair and which is biased, but it would not be full-proof as random events have all kinds of possible variability. – Chaotic Good Feb 22 '22 at 21:27
  • If you have an a priori notion of what the probabilities should be, you can use Bayes' Theorem to hone your estimates. But if you have no information at all, empirical data is all you have. And yes, as you surmise, it is almost surely wrong. the more trials you run, the better the empirical estimate should be. – lulu Feb 22 '22 at 21:27
  • The reason I mention the number of trials is "reliability". We know that the more trials we do, the more reliable the result is. For example, if you flip a fair coin 10000 times, you will see the probability of it landing on heads is close to 50%. – jun Feb 22 '22 at 21:29
  • For your second question, you might consider a Chi-squared test. – awkward Feb 23 '22 at 16:13

2 Answers2

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  1. 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.

  2. 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.

Nuclear Hoagie
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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.

Brady Gilg
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  • You can never know (with certainty) the true distribution after an infinite number of trials either, even if carrying out an infinite number of trials were possible (which it doesn't seem to be in reality)... – Adam Rubinson Feb 22 '22 at 21:45
  • Thanks, this is what I needed! – jun Feb 22 '22 at 21:49