# Probability of getting extra heads

Suppose you are tossing a coin $10$ times. You observe that the first $5$ tosses result in all heads. Then what is the probability that you would get $3$ heads in the remaining $5$ tosses? I have no idea about how to solve the problem...please help. What I think is that we have to recalculate the biasing of the coin from the data given using Bayes' theorem.

• Is the coin assumed fair? If so, the outcome of the first five tosses has no effect on the second five. – Matthew Leingang May 28 '15 at 17:38
• There is no "correct" answer. You need to "infer" something, and there could be different methods that will give different results. – grdgfgr May 28 '15 at 17:38
• I came across the question while solving a book so it is not wrong and there is only one answer...also isn't it quite obvious that the coin is not fair.. – Abhishek Bakshi May 28 '15 at 17:43
• In order to solve the problem using Bayesian methods, we need a prior distribution for the random variable $P$ that gives the probability of head. No prior has been specified. – André Nicolas May 28 '15 at 17:57

Unless you have some particularly good reason to suspect trickery (e.g., it is a magician who is flipping the coin, and maybe there are heads on both sides), you can assume the coin is fair. In that case, the first five tosses are irrelevant. It isn't surprising at all to see $5$ heads in a row—I'd get suspicious at seeing $10$ in a row, maybe.

The probability of getting $3$ heads in $5$ flips is $\dfrac{5 \choose 3}{2^5} = \dfrac{10}{32} = \dfrac{5}{16}$.

Edit: What if we're told in advance that the coin is biased? Well, we still don't have enough information to make a conclusion. Is it a coin with heads on both sides, so that every flip is a head? Is it ever so slightly unbalanced, so it lands heads $50.00001 \%$ of the time? Or maybe it favour tails, but we just got lucky with $5$ heads in a row? Without more information, we can't make a meaningful conclusion.

In short, if we were told that the coin is biased, then I would recommend: (1) physically inspecting the coin; and (2) flipping it $1000$ times in a row and seeing what happens.

• Couldn't the coin have been a biased one?? what if it was given in the question that the coin is biased – Abhishek Bakshi May 28 '15 at 17:54
• @AbhishekBakshi If it was given in the question that the coin is biased, then you do have a particularly good reason to suspect trickery. If it's not given in the question, then I would assume that the coin is just a coin. – Théophile May 28 '15 at 17:58
• Ok, forget about this question...what answer would you have given if the in the question it was said that the coin was biased...please put this answer in your solution too.. – Abhishek Bakshi May 28 '15 at 18:01
• @AbhishekBakshi Sure; I've updated my answer. – Théophile May 28 '15 at 20:11
• Well, I'm satisfied with your answer...let's just say that prior to the coin flipping, another such experiment was done and it was found that we got $8$ heads in $10$ flips. Then are we allowed to assume that the coin is biased and the probability of getting heads in a toss is $\frac 8{10}$ and solve the question according to that. What would be the answer in that case? – Abhishek Bakshi May 29 '15 at 12:42