Question 4.6: Product Failure Study A product consists of two components. ......
Product Failure Study
A product consists of two components. The product fails when either or both components fail. There is a 5% chance that component 1 will fail and a 10% chance that component 2 will fail. The components can fail independently of each other.
(See Excel file C4.3 – product failure.)
What is the probability that the product will fail (i.e. that either or both components will fail together)?
This problem can be represented in a probability tree as shown in Figure 4.6.
The first two branches represent the two outcomes of component 1 (i.e. \text F_1 = component 1 fails and \text S_1 = component 1 does not fail). The second set of branches represents the two outcomes of component 2 (i.e. \text F_2 = component 2 fails and \text S_2 = component 2 does not fail).
The end of each branch represents the joint event of outcomes for both components together. For example, the top branch represents component 1 failing and component 2 failing. Therefore the joint probability of both components failing is found by multiplying their marginal probabilities (i.e. applying Formula 4.6 – the multiplication rule for statistically independent events). The application of Formula 4.6 is repeated for each branch of the tree.
Since the product will fail when either component 1 fails (and not component 2) or component 2 fails (and not component 1) or both components together (component 1 and component 2) fail.
Thus P(product fails) = \text P(\text F_1\cap \text S_2)+\text P(\text S_1\cap \text F_2)+\text P(\text F_1\cap \text F_2).
These joint probabilities can be read off the end of each appropriate branch in the probability tree.
P(product fails) = 0.045 + 0.095 + 0.005 = 0.145
Thus there is a 14.5% chance that this product will fail.
