Final answer:
The question involves finding the probability of a machine working based on the probability of its components working. Using the rules of probability, we calculate the likelihood of components A, B, and C functioning, to ultimately determine the probability of the machine working, which depends solely on component C.
Step-by-step explanation:
The problem given discusses the functionality of a machine based on its components A, B, and C. The machine works only if component C is working. We start by finding the probability of each component working, and then use those probabilities to find the likelihood of the machine working as a whole. We apply basic rules of probability such as the multiplication rule (P(A AND B) = P(A|B)P(B)), and conditional probability (P(A|B) is the probability of A given B has occurred).
The probability of component A working is given as 7/10. This is our starting point. Next, regardless of whether A is working or not, the probability of B working is 1/3. The tricky part comes with component C: if both A and B are working, the probability of C working is 5/6; otherwise, it is just 1/10. To compute the probability of the entire machine working, we must consider the probability of A and B both working, which then influences the probability of C.
Let's go through the steps:
- Calculate the probability of A and B both working:
P(A and B) = P(B|A)P(A) = (1/3)(7/10) = 7/30. - Calculate the probability of C working given A and B are both working, and then find the overall probability of C working based on A and B's status. This involves considering the two scenarios: A and B are both working or at least one of them is not working.
- Finally, since the machine works only if component C is working, the probability of the machine working is the same as the probability of C working in the 'best-case' scenario (A and B both working).
Once we have calculated the probabilities of these events, we can know the likelihood of the machine being operational.