$\textbf{The Problem:}$ I have three coins in my pocket, labeled $a,b$ and $c$. The probabilities with which the coins give tails are $1/2,1/3,3/4,$ respectively. I grab one coin out of the three at random and I flip it twice.
Define a probability space that models the choice of coin and the outcomes of the coin flips.
$\textbf{Thoughts and Concerns:}$ For the sample space I have the following somewhat obvious model $$\Omega=\{(x,y_1,y_2)\,:\,x\in\{a,b,c\}\text{ and }y_i\in\{H,T\}\}.$$ Upon noticing that $\Omega$ has finite cardinality, we see that we may take our $\sigma$-algebra to be $\mathcal F=2^\Omega.$
I am having difficulties when it comes to defining a suitable probability measure.
Could anyone point me in the right direction?
Any feedback is much appreciated, and I want to thank you for your time.
You only have $12$ points in the probability space, so you just have to specify the probability of each of them. For example, $(b,T,H)$ should have probability $$\frac13\cdot\frac13\cdot\frac23=\frac{2}{27}$$ Just do the same kind of thing for the other $11$ points.