91Ó°ÊÓ

To calculate the probability of the entire system functioning, we need to find the complement - the probability that none of the components are working. Since each component works with probability \(\frac{1}{2}\), it also fails with probability \(\frac{1}{2}\). The probability of all components failing, \(P\left( \bigcap_{i=1}^n \text{component }i\text{ fails} \right)\), considering that they work independently: \(P\left( \bigcap_{i=1}^n \text{component }i\text{ fails} \right) = \left(\frac{1}{2}\right)^n\) Now, we can find the probability of the entire system functioning, which is the complement of the probability of all components failing: \(P(B) = 1 - P\left( \bigcap_{i=1}^n \text{component }i\text{ fails} \right) = 1 - \left(\frac{1}{2}\right)^n\) #Step 2: Find the probability of component 1 working and the system functioning#

In this step, we need to calculate the probability of component 1 working, given that the system is functioning. We'll first consider the probability both component 1 is working and the system is functioning: \(P(A \cap B) = P(\text{component 1 works} \cap \text{system functions})\) Since the system is functioning if and only if at least one component is working, the system is also functioning if component 1 is working. Therefore, we know: \(P(A \cap B) = P (\text{component 1 works}) = \frac{1}{2}\) #Step 3: Calculate the conditional probability# Now we can apply the definition of conditional probability, \(P(A|B) = \frac{P(A \cap B)}{P(B)}\):

\(P(A|B) = \frac{P(\text{component 1 works} \cap \text{system functions})}{P(\text{system functions})} = \frac{\frac{1}{2}}{1 - \left(\frac{1}{2}\right)^n} = \frac{1}{2 - \left(\frac{1}{2}\right)^{n-1}}\) Thus, the conditional probability that component 1 works given that the system is functioning is \(\frac{1}{2 - \left(\frac{1}{2}\right)^{n-1}}\).