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You have two coins, one is biased heads and the other biased tails. Someone picks a coin without you knowing which one, then starts flipping. The first toss s

by j7ake 3y ago

You have two coins, one is biased heads and the other biased tails. Someone picks a coin without you knowing which one, then starts flipping. The first toss shows heads. Does that mean subsequent flips from the same coin will be more likely to be heads? Yes. Answer is only no if all coins are the same with no bias, which they are not.

jncfhnb 3y ago
I’m not clear why you posted this. This is a different problem from the one being discussed.
- j7ake 3y ago
  The problem being discussed is the same as my toy example, just extended to more coins. You now have 101 coins, with biases ranging from 100 percent heads (or urn with only red balls) to 100 percent tails (or urn with only green). Someone chooses one of the 101 coins randomly (each coin has equal probability of being chosen) and starts flipping it. First flip shows heads, what is the probability the next flip is also heads? Answer is 2/3. The point is that knowing what was the first flip gives information about what biases a coin may have. The original problem has a minor technical twist that balls are drawn without replacement, but it’s an irrelevant detail for N>10 total balls.
  - jncfhnb 3y ago
    I mean, sure, what you’re describing is relevant to the original discussion, but you’ve posted it in a sub thread about how it changes if it is a 50/50 binomial distribution. You’ve replaced it with a binomial of a uniformly selected p, which ends up being, amusingly, the same as a uniform.
    - j7ake 3y ago
      As long as you think the flipper chose their coin from a bag of coins with varied biases, then first flip always gives actionable information. It’s mostly irrelevant what kind of distribution this bag of coins comes from (except for some degenerate cases). What matters is that selecting coins from this bag gives coins of varying biases.
      - jncfhnb 3y ago
        Yes but the question being discussed assumes they are fair. That’s why it’s a different question.

by j7ake 3y ago

jncfhnb 3y ago

I’m not clear why you posted this. This is a different problem from the one being discussed.

j7ake 3y ago
The problem being discussed is the same as my toy example, just extended to more coins. You now have 101 coins, with biases ranging from 100 percent heads (or urn with only red balls) to 100 percent tails (or urn with only green). Someone chooses one of the 101 coins randomly (each coin has equal probability of being chosen) and starts flipping it. First flip shows heads, what is the probability the next flip is also heads? Answer is 2/3. The point is that knowing what was the first flip gives information about what biases a coin may have. The original problem has a minor technical twist that balls are drawn without replacement, but it’s an irrelevant detail for N>10 total balls.
- jncfhnb 3y ago
  I mean, sure, what you’re describing is relevant to the original discussion, but you’ve posted it in a sub thread about how it changes if it is a 50/50 binomial distribution. You’ve replaced it with a binomial of a uniformly selected p, which ends up being, amusingly, the same as a uniform.
  - j7ake 3y ago
    As long as you think the flipper chose their coin from a bag of coins with varied biases, then first flip always gives actionable information. It’s mostly irrelevant what kind of distribution this bag of coins comes from (except for some degenerate cases). What matters is that selecting coins from this bag gives coins of varying biases.
    - jncfhnb 3y ago
      Yes but the question being discussed assumes they are fair. That’s why it’s a different question.