I am solving my homework of Probability class. I am not sure my approaches are correct.
Questions:
Three switches connected in parallel operate independently. Each switch remains closed with probability $p$.
- What is the probability of receiving an input signal at the output?
- What is the probability that one switch is open given that an input signal is received at the output?
For Q1, I approached by complement event. 'Probability of at least one switch is closed' = 1 - 'Probability of all switches are open' => $1-(1-p)^3 = p^3-3p^2+3p = p(p^2-3p+3)$
For Q2, I approached by Bayes' theorem. Event X = { Only one switch open }, Event Y = { At least one switch is closed }
$P(X) = 3p^2(1-p)$
$P(Y) = p(p^2-3p+3)$ ... This from the result of Q1.
$P(X \cap Y) = P(X)$ ... This is because if two switches are closed, output always receives input signal (that means X is subset of Y)
=> $P(X|Y) = P(X \cap Y)/P(Y) = P(X)/P(Y) = 3p^2(1-p)/(p(p^2-3p+3)) = 3p(1-p)/(p^2-3p+3)$
Could you check I am going right direction?