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I understand that independent events (such as a fair coin flip) should not be viewed in succession. For example, if you flip heads 10 times in a row, the odds of flipping the next coin heads is still 50%.

However, there is another way of looking at the coin flips. What is the probability of flipping 11 heads in a row And I already know that to be 0.5^11 ~= 0.1%

So if you were making a bet on coin flips and 10 heads came up, would you still base a bet on the 50% fact, or on the odds of getting heads 11 times in a row (.1%) fact? If you consider the 50% fact or 0.1% fact, please explain why?

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The 10 heads coming up are irrelevant to the outcome of the 11th coin flip. Once you see those first 10 flips, they have been determined and there is no randomness from those events. So, you still have a 50% chance of getting heads on the 11th flip.

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This really depends on how sure you are that the coin is really a fair coin. I'm not sure if you're familiar with the basics in Bayesian statistics, but you would start with a prior distribution for the true proportion of heads that reflects your belief ("probability") of the fairness of the coin and then update that prior distribution into a posterior distribution. For example, in this situation, you might typically have a $Beta(10,10)$ distribution as your prior. Having seen $10$ heads in a row, you would update it to be a $Beta(20,10)$ distribution, and then make inference based on that distribution. For example, your prior probability for the proportion of heads being greater than $0.5$ was exactly $0.5$ using the $Beta(10,10)$ density; Your posterior or updated probability, now using the $Beta(20,10)$ would then be about $0.97$.

But again, this all depends on how sure you are to start with; if you know irrefutably that the coin is fair, then of course the probability remains at $0.5$.

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  • $\begingroup$ I plan to make a follow up question to this related to sports. Say you have a player (Lebron James) who is known to be 50% from the field. If he misses 10 shots in a row, could you assume he could make his next shot? Or can you make no assumptions? Or.. can you update prior distribution into a posterior distribution $\endgroup$
    – E.S.
    May 30, 2015 at 1:17
  • $\begingroup$ In that case you can still update a prior, yes; however, since you're basing that 50% reliability on all of his previous shots, let's say that those sum up to 1000, then your prior belief is a lot stronger - something like a $Beta(500,500)$ might not be unreasonable - meaning updating it will have less of a noticeable effect - the posterior would be a $Beta(500,510)$, which is hardly different. $\endgroup$
    – Brent
    May 30, 2015 at 1:47
  • $\begingroup$ Hmm I don't recall that terminology from my college stat classes. I wonder if what you wrote could be written in more layman terms? $\endgroup$
    – E.S.
    May 30, 2015 at 5:06

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