Let us consider the following question
Devasena can select either Baahubali or Bhallaladeva but not both.
If she selects Baahubali, then she will be happy with a probability $\dfrac{9}{10}$;
If she selects Bhallaladeva then she will be happy with a probability $\dfrac{1}{10}$.
Devasena decides to select one among them based on tossing a fair coin.
What is the probability that she will be happy with Bhallaladeva?
I know that the answer is 1/20 and how to solve it using conditional probability;
But I want to represent every event and sample space in this problem as explicit sets (either discrete or continuous), either using roster form or set builder form and calculate probability according to it. demystify.
Let me know if anyone sees a way to simplify or improve the following solution:
We could let the sample space be $$ S = \{(\text{Baahubali},\text{Happy}), (\text{Baahubali},\text{Unhappy}), (\text{Bhallaladeva},\text{Happy}),(\text{Bhallaladeva},\text{Unhappy})\}. $$ If $E_1$ is the event that she selects Baahubali, and $E_2$ is the event that she selects Bhallaladeva, then we have $$ E_1 = \{(\text{Baahubali},\text{Happy}), (\text{Baahubali},\text{Unhappy})\} $$ and $$ E_2 = \{(\text{Bhallaladeva},\text{Happy}),(\text{Bhallaladeva},\text{Unhappy})\}. $$ If $F_1$ is the event that she is happy and $F_2$ is the event that she is unhappy, then we have $$ F_1 = \{(\text{Baahubali},\text{Happy}),(\text{Bhallaladeva},\text{Happy})\} $$ and $$ F_2 = \{(\text{Baahubali},\text{Unhappy}), (\text{Bhallaladeva},\text{Unhappy}). $$
We make the modeling assumptions that $P(E_1) = P(E_2) = 1/2$ and also $$ P(F_1 \mid E_1) = \frac{9}{10}, \qquad P(F_1 \mid E_2) = \frac{1}{10}. $$ Finally, $$ P(E_2 \text{ and } F_1) = P(E_2) P(F_1 \mid E_2) = \frac12 \cdot \frac{1}{10} = \frac{1}{20}. $$