Probability of Independent Events individual vs in series

Question

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?

Answer 1

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.

Answer 2

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 $0.5$.