The image you've uploaded seems to show a problem related to electrical circuits, with some probability and logic considerations.

The problem presents a series-parallel electrical circuit with three components $A$, $B$, and $C$. The text mentions that these components are working or failing independently, with the same probability of failure for each, denoted by $p$.

Problem Breakdown:

• $P(A) = P(B) = P(C) = p$: The probability of failure for each of the components is $p$.
• The components $A$, $B$, and $C$ are arranged in a combination of series and parallel circuits.
• You are asked to find the probability that the circuit functions when it is known that component $A$ is working.

Circuit Configuration:

1. $A$ and $B$ are in parallel, meaning the circuit works as long as one of these two components is functioning.
2. Component $C$ is in series with the $A$-$B$ parallel combination. This means the circuit will only function if $C$ is also functioning.

Conditional Probability:

The question likely asks for the probability that the entire circuit functions, given that $A$ is functioning.

Solution:

1. Parallel section (A and B):
   The probability that the parallel section functions given that $A$ is working is: $P(\text{Parallel works} | A \text{ works}) = 1$ Since $A$ is working, the parallel section will always work, because at least one component (A) in the parallel combination is functioning.

2. Series with $C$:
   The circuit will work only if $C$ is also working. The probability that $C$ is working is: $P(C \text{ works}) = 1 - p$ Therefore, the probability that the circuit functions, given that $A$ is working, is just the probability that $C$ is working: $P(\text{Circuit works} | A \text{ works}) = P(C \text{ works}) = 1 - p$

Thus, the probability that the entire circuit works given that $A$ is functioning is $1 - p$.

Would you like further details or clarifications?

Related Questions:

1. How would the result change if component $B$ was guaranteed to be functioning instead of $A$?
2. What is the probability that the circuit functions if none of the components' states are known?
3. How does the overall reliability of the circuit change if the probability of failure for each component increases?
4. If the components $A$, $B$, and $C$ had different probabilities of failure, how would you approach the problem?
5. What is the probability that all components are functioning?

Tip:

When dealing with series and parallel circuits, remember that series circuits fail if any component fails, while parallel circuits can continue to operate as long as at least one component works.