Coin tossing, probability and some logic, kinda

Swan Red · October 3, 2014

what are the odds it lands tails next time?

Most seem to argue that the probability of the coin landing tails is unconditional on previous coin tosses. I think this is a mistake, I've mentioned this before and Paul thinks it's silly and John has also referred to it.

My point is that the probability of it landing tails is dependent on the assumption that the coin is a fair coin. A fair coin being defined as one that is as likely to land heads as tails. What confidence should we have that this assumption is correct? It seems to me entirely feasible that some very small % of coins may though accident or design have some tendency towards heads or tails. If this is correct then we should consider the probability conditional.

If we toss a coin 10 times and it lands heads it seems we can't do worse by backing heads last time, if indeed some coins are biased then we should include that in our calculations and weight the probability accordingly.

We don't need 10 tosses for this though, if we employ Bayes Theorem we should our assessment of the probability should be modified every time we know something new, like how the coin landed last time.

Consider then the prior probabilities of two coin tosses, the standard response would be that the Probability of the 4 possible outcomes HH = HT = TH = TT = .25. The Bayesian disagrees, the Bayesian considers the following P(H|H) = P(T|T) > P(H|T) = P(T|H)

Why is the Bayesian wrong?

surf · October 3, 2014

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