Wordy Bayesian Question Five million boys below the age of five live in Erewon. The priests of Erewon are sure that one of them (chosen by fate at random) embodies the spirit of Captain Coin Tosser, who can predict heads or tails of a flipped fair coin with perfect accuracy. In order to find this boy, they travel the country giving a test: each boy being tested is asked to predict which way a flipped coin will land, eighteen times.
After months of grueling search, they find a boy who passes their test, correctly predicting all eighteen tosses. Taking the priests' belief as a given, what is the probability that this boy is in fact the embodiment of the spirit of Captain Coin Tosser?  
So:
$$\mathbb{P}(C) = \frac{1}{5,000,00},\;    \mathbb{P}(T)= \frac{1}{(0.5)^{18}}$$
And we want $\mathbb{P}(C\mid T) = \frac{\mathbb{P}(T\mid C)\mathbb{P}(C)}{\mathbb{P}(T)}$ but we don't have $\mathbb{P}(T\mid C)$ from what I can see.

The calculation we want is as follows:
$$
\mathbb{P}(C \mid T) = 
\frac{\mathbb{P}(T\mid C)\mathbb{P}(C)}
{\mathbb{P}(T\mid C)\mathbb{P}(C) + \mathbb{P}(T\mid \text{not }C)\mathbb{P}(\text{not }C)}
$$
We have 


*

*$\mathbb{P}(T\mid C) = 1$

*$\mathbb{P}(T\mid \text{not }C) = \frac{1}{2^{18}}$

*$\mathbb{P}(C) = \frac{1}{5,000,000}$

*$\mathbb{P}(\text{not }C) = 1 - \frac{1}{5,000,000}$


For the sake of efficient computation, we can rewrite that as 
$$
\mathbb{P}(C \mid T) = \left(1 + \frac{\mathbb{P}(T\mid \text{not }C)\mathbb{P}(\text{not }C)}{\mathbb{P}(T\mid C)\mathbb{P}(C)} \right)^{-1}
$$
You should get a probability of approximately $5\%$.
