How to exclude combinations for defined sequences?

In summary, the conversation revolves around a probability problem regarding a factory's product failures and the likelihood of having to close its production line. The problem becomes increasingly complex as the time frame expands, and the speaker is seeking help with finding a general rule to solve the problem for a 10-year period. The suggested approach involves using markov chains or a renewal model, but the speaker is unsure due to their limited math background. They are open to simpler solutions and are seeking assistance in solving the problem.
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Josefk
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Please can anyone help with the below problem? It’s an interesting problem but please bear with me as i don't have much math background.

A factory’s product is sampled once per month every month by its quality inspection team. The factory is allowed up to 2 product failures per ROLLING 12 month period (i.e. Mar-Feb, April-Mar etc) but if it fails 3 times it must close its production line. The probability of failing each sample is 0.05. Work out the probability that (given that the factory hasnt had any failures in the last 12 months) the factory will have to close its production line on AT LEAST one occasion:
a) in the next 12 months
b) in the next 13 months
c) in the next 15 months
d) in the next 10 years
********************
My answers so far:
I really need help with part c) and particularly part d).
a) using standard binomial probability (12!/3!9! x 0.05^3 x 0.95^9)+(12!/4!8! x 0.05^4 x 0.95^8)+ etc ... +(12!/12!0! x 0.05^12 x 0.95^0) = 0.0196.
b) Using the number of combinations taken from part a), i deducted from the first term those combinations that do not have 3 failures within 12 months of each other (let’s call this the ‘no shutdown’ combinations). To work this out, i reasoned the only possible combinations are those with a single failure at both ends of the range. So i did 2!/2!0! X 11!/11!0! = 11. (this seems correct after i physically sketched the combinations out!). There is no need to repeat this for the second term (for 4 failures) as there is no possible combination here that won't have 3 failures within 12 months of each other. The overall probability i calculated to be 0.0237.
c). I’m struggling to work out the number of ‘no shutdown’ combinations for 3 failures within 15 months and also for 4 failures (as this term now becomes significant). I need to be able to develop a rule for working this out. Similar to part b) i have tried to work out the combinations of failures occurring within the end 4 months of the range (i.e. months 1,2,14,15) such to avoid 3 occurring within 12 months. To do this i did (4!/3!1! X 11!/11!0!) + (4!/2!2! X 11!/10!1!) + (4!/1!3! X 11!/10!2!) = 92. However, when i sketched the combinations out by hand there are only 76, so i seem to be double counting 16 combinations using the above approach.

In order to work out part d) i will need a general rule to exclude the ‘no shutdown’ months for any number of months (m) being considered and any number of failures up to m-3.
CAN ANYONE HELP PLEASE?
 
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  • #2
Josefk said:
Please can anyone help with the below problem? It’s an interesting problem but please bear with me as i don't have much math background...

d) in the next 10 years...

In order to work out part d) i will need a general rule to exclude the ‘no shutdown’ months for any number of months (m) being considered and any number of failures up to m-3.
This could be an issue. Structurally, this feels like a modified runs problem, which means it is fairly easily addressable with markov chains or a renewal model. Doing the 10 year problem with raw combinatorics may be technically possible with inclusion-exclusion, though it seems laborious and unpleasant.

So, what do you know and where is this problem coming from? If you really don't have much of a math background, I'd suggest working out all the mechanics on a, b, and c and ignoring d.edit: I may be able to frame the renewal argument as just a big linear recurrence involving mutually exclusive events -- if you're ok with such things then that should work.
 
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FAQ: How to exclude combinations for defined sequences?

What is the purpose of excluding combinations for defined sequences?

The purpose of excluding combinations for defined sequences is to limit the number of possible outcomes in a scientific experiment or study. By excluding certain combinations, researchers can focus on specific variables and better understand their effects on the overall outcome.

How do you determine which combinations to exclude for defined sequences?

The process of determining which combinations to exclude for defined sequences involves careful analysis and consideration of the variables involved in the experiment. Researchers must identify which combinations are not relevant to the study or may skew the results, and then exclude them accordingly.

Can excluding combinations for defined sequences affect the validity of the study?

Yes, excluding combinations for defined sequences can potentially affect the validity of a study. If the excluded combinations are important factors in the outcome of the study, the results may not accurately reflect the real-world scenario. It is important for researchers to carefully consider which combinations to exclude and justify their decisions.

Are there any alternative methods to excluding combinations for defined sequences?

Yes, there are alternative methods to excluding combinations for defined sequences. One method is to use statistical techniques such as randomization or blocking to account for all possible combinations. Another method is to conduct a sensitivity analysis, which involves testing the impact of including or excluding certain combinations on the overall results.

What are the potential limitations of excluding combinations for defined sequences?

One potential limitation of excluding combinations for defined sequences is that it may not accurately reflect real-world situations. By limiting the combinations, the study may not account for all possible scenarios and may not be generalizable to a larger population. Additionally, excluding combinations may also introduce bias into the study if the excluded combinations are important factors in the outcome.

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