# Interpreting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by unfair I mean, the probability of head is 2/3 instead of 1/2. Now, imagine a scenario where I toss this coin 10 times and I get 10 heads. So if I want to get the posterior, I do the following:

$$P(\texttt{unfair}|\texttt{evidence}) = \frac{P(\texttt{evidence}|\texttt{unfair})*P(\texttt{unfair})} {P(\texttt{evidence})} = \frac{P(\texttt{evidence}|\texttt{unfair})*P(\texttt{unfair})} {P(\texttt{evidence}|\texttt{unfair})*P(\texttt{unfair})+P(\texttt{evidence}|\texttt{fair})*P(\texttt{fair})} =\\ \frac{(2/3)^{10}*(0.1)}{(2/3)^{10}*(0.1)+(1/2)^{10}*(0.9)} = \frac{0.0017}{0.0017+0.0009} = \frac{0.0017}{0.0026} = 0.653$$

Now how can I interpret this posterior? It's technically saying that the probability of the coin being unfair given the evidence is higher than the prior belief on the coin being unfair (0.653 > 0.1), but is still below the prior belief on the coin being fair (0.653 < 0.9). So, the only conclusion is, the coin is still more likely to be fair. So, in order to change our belief about the coin we have "being unfair", we need more evidence (i.e. 100 heads in a row). Is it correct?

• Note that enclosing LATEX inside of dollar signs $renders it as math. For example: \alpha is$\alpha$. – Sycorax Commented Apr 27, 2015 at 2:51 • For$n$tosses of Heads to make your posterior suggest an unfair coin is now more likely than your initial prior probability for a fair coin of$0.9$, you need$\dfrac{(\frac23)^n}{(\frac12)^n} >(\frac{0.9}{0.1})^2$i.e.$n > \dfrac{2\log(9)}{\log(\frac43)} \approx 15.3$so$n \ge 16$. This is unlikely even if the coin is unfair in way described:$(\frac 23)^{16}\approx 0.0015\$ Commented Sep 5, 2023 at 12:49

• $$\frac{0.0009}{0.0026} = 0.34$$ which is lower than 0.653. Based on the updated posterior for fairness, we can conclude the coin is more likely to be unfair? Commented Apr 27, 2015 at 4:28