Bayesian Probability

Question

Dave flips a coin with a 60% (40%) probability of landing heads (tails). If the coin lands heads, Kevin chooses a random variable $X$ from a uniform probability distribution $u(x)$ in the range $-1 \leq X \leq 1$. If the coin lands tails, Kevin chooses $X$ from the same range but from a probability density of $p(x)=\frac{1}{2}+\frac{x}{5}$. Kevin tells you $X=\frac{1}{2}$. 


What probability can you assign to Dave's coin having landed heads?

Answer

$P(Heads|X=\frac{1}{2})=\frac{P(X=1/2|Heads)*P(Heads)}{P(X=1/2)}$

$P(Heads|X=\frac{1}{2})=\frac{3/5*u(1/2)}{3/5*u(1/2)+2/5*p(1/2)}$
$P(Heads|X=\frac{1}{2})=\frac{3/5*1/2}{3/5*1/2+2/5*3/5}$
$P(Heads|X=\frac{1}{2})=\frac{5}{9}$