# Probability of Independent Events individual vs in series

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?

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 $0.5$.
• 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. – Brent May 30 '15 at 1:47