Stanton D. answered • 05/19/20
Tutor to Pique Your Sciences Interest
Hi Shanil P.,
This problem seems formidable only because one normally considers the failure probability of a single battery (usually when you are stuck on the road in the middle of nowhere!), and that as a function of service time. However, consider the batteries to be of type A (fails by 70 months) or B (still functions at 70 months). Then, you are simply picking A's and B's at random from an infinite pool, until you have 100 of them, and taking the proportion. That's a binomial probability, and you can find calculators (and formulas) for it online. Curiously enough, the sticking point may be the phrase " 'within' 0.05". Does this mean inclusive, i.e. are we talking 75-85 B batteries, or exclusive, i.e. we are talking 76-84 B batteries? If inclusive, failure to meet is 0.08747538464+0.08044372114 or 0.16791910578 ; if exclusive, failure to meet is 0.12850551484+ 0.13135321733 or 0.25985873217 . So those would correspond to "meets" probabilities of 0.832 or 0.740 for those inclusive or exclusive ranges respectively. That's quite a difference, and you might discuss with your instructor what his/her interpretation of "within" is. Personally, I take within to be inclusive, but that is relative to a career as an analytical chemist in a legally regulated industry, in which "not more than (after rounding as specified)" is the tolerance-permitted language.
One thing this problem should (perhaps) teach you is, don't go trying to calculate things you shouldn't, such as here a standard deviation! Tempting, perhaps, given that you can postulate 4-B batteries + 1-A battery as a calculation unit -- but useless. Standard deviation is only for continuous variables, not discrete ones!
-- Cheers, -- Mr. d.
