Given two lottery tickets, out of which one can be bought. The first lottery \$100 can be won with probability 0.1, and the price of ticket is \$10. In the second lottery, \$50 can be won with probability 0.1 and \$500 with probability 0.01. The price of ticket is \$20. To decide which ticket to buy a fair coin can be tossed. In case of head, first ticket is chosen else the second ticket. Let $X$ be random variable that denotes the net payout (taking into account price of a ticket). Find probability mass function of $X$?
I tried doing the above by using law of total probability by using the conditional probability $P(X|H)$ and $P(X|T)$ and the respective probabilities $P(H)=\frac{1}{2}$ and $P(T)=\frac{1}{2}$. So here $P(X)=P(X|H)*P(H)+P(X|T)*P(T)$. Here I assumed $P(X|H)=p$ and $P(X|T)=q$. So here $\Sigma p=1$ and $\Sigma q=1$ for the first and the second lottery ticket.
Would the expected value of $X$ be dependent on different payout or probabilities? In the above case, it is coming as average of the expected values of net payouts for each of two lotteries.
What are the possible discrete values that random variable $X$ (net payout) can take on and what are their probabilities? For example, one can have net payout of $30$ with probability $\frac{1}{2} \times 0.1 = 0.05$. The same is true for net payout of $90$.
$p(x)$ = $\left \{\begin{array} {l}0.05 \ \ \ \ \ \ \ \ x \in \{30, 90\} \\ ... \\ ... \end{array} \right.$
Can you complete this by finding other discrete values of $X$ and associated probabilities?