At a school exam, we have a question as follow (tech-active exam, graphic calculator is allowed):
A manufacturer produces tins of sardines; the tins are sold in boxes of two. The manufacturer runs many production batches each day at its factory. The manufacturer recently learned that its tins of sardines may be contaminated with trace amounts of toxin A. However, the toxin has not yet been detected in any production batch. To test for contamination, one box of sardines is taken aside for testing at the beginning of each production batch. Both tins from the box are tested for traces of toxin A as well as other contaminants. If either tin is found to be contaminated, the entire production line will be shut down. For each production batch, the probability that a tin of sardines is contaminated is normally distributed with a mean of 0.25 and a standard deviation of 0.05. The following information is also assumed.
• The only possible contaminant is toxin A.
• The contaminant test is 100% accurate. A production batch is about to begin. Determine the probability that the production line will have a less than 30% chance of being shut down.
A production batch is about to begin.
Determine the probability that the production line will have a less than 30% chance of being shut down.
The solution given by the school is shown in this image: The school solution says the probability will be approximately 4%
However, I'm not sure if it is correct as it kind of assumed the probability that a tin of sardines is contaminated for both tins is the same, while they should be indepent. For example, I do not think the solution have considered the case where the first tin have 29% of chance while the second tin have 1% of chance of being contaminated. Therefore, I think the actual solution should be lower than the 4% given by the school.
This is my attempt on the question (likely wrong though), I got 0.67%
I would like to know whether the school solution is correct or not.
Big hat off to whoever that can help me with this question, I'm really confused (Sorry for bad grammar, I'm not native to the language)