Answer:
The maximum probability for each of the components in a system to fail, for you to still achieve your objective, is 0.008992
Step-by-step explanation:
- There are 1,000 different components in each system.
- The Objective:
Each system should have a 0.999 probability of functioning properly. In other words, your department (or you) expects that each system will have a high probability of functioning properly, hence the closeness of 0.999 to 1.
- Each system is such that at most 8 components out of 1,000 can be bad, for it to still function.
- Question:
What must be the maximum probability for each component in a system to fail, for the system to still have a 0.999 probability of functioning properly?
ANSWER:
When the question says maximum, you need to assume that for each system, 8 components are actually bad!
1000 - 8 = 992
So with 992 components (at least or minimum) functioning, a system will still function.
Next Step:
Since the objective (the expected probability) for a system to function is 0.999, the probability of each of the minimum 992 components to function (which is same as the maximum probability for each of the 1,000 components to function) is
992/1000 × 0.999
=0.992 × 0.999
=0.991008
Now, remember that the question says fail, not succeed. So, if the figure above is the maximum success rate for each of the 1,000 component parts (same as the success rate for each of the minimum of 992 component parts), then the failure rate or probability is (1 - 0.991008 = 0.008992).
FINAL ANSWER:
The maximum probability for each of the components in a system to fail, for you to still achieve your objective, is 0.008992
