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Every time Sherry buys a book it is a math book with probability 0.3 and a novel with probability 0.7. Sherry keeps her math books in a bookcase that can hold 50 math books and keeps her novels in a different bookcase that can hold 100 novels. What is the probability that she fills up her math bookcase before she fills up her novels bookcase, assuming they are initially empty? Assume Sherry continues her book buying patterns even after one or both of her bookcases gets filled. Use indicator random variables and the central limit theorem to approximate the desired probability.

Question: Is be attempt below correct?

My attempt:

Let p = 0.3, n = 149 purchases

$P$[she fills up at least 50 math books in 149 purchases] = $P$[($X$ - np)/ √(np(1-p))] = $P$[($X$ - (149)(0.3))/√(149(0.3)(0.7)) ≥ (50 - 44.7)/√(31.29)] = $P$[$Z$ ≥ 0.95] = $0.8289$

Therefore, the probability of filling up her math bookcase before her novel bookcase is approximately 83%.

queence
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  • This question is identical to the one at https://stats.stackexchange.com/questions/326828/competing-negative-binomials/327015#327015, only with different numbers. – whuber Mar 28 '18 at 18:19
  • Thank you, but I am looking to use the Central Limit Theorem to solve it. – queence Mar 28 '18 at 18:24
  • The CLT does not apply. At best you could use the Normal approximation to the Binomial distribution--and apply exactly the same solution I referred to. – whuber Mar 28 '18 at 18:27
  • okay, I'll try that. The rest of the question posted above asks for you to solve it via central limit theorem, so that's why I attempted it. – queence Mar 28 '18 at 18:34
  • If we show the people that gave you this a statement of the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem) and ask what they actually mean for us to do with that theorem (where in this question do we have $n\to\infty$?), it will turn out that they actually mean for you to use the normal approximation to the binomial, just as whuber was suggesting. That approximation can apply in some circumstances even at quite small $n$, not just in the limit as $n\to\infty$, but the suitability of such small sample approximations is not actually in the purview of the CLT... ctd – Glen_b Mar 29 '18 at 01:03
  • ctd ... You might be able to invoke the Berry-Esseen inequality to bound the error (in a small-n approximation); the more recent versions of the inequality often indicate that the approximate answer must be quite accurate (though by the time you check such a bound you could do an exact binomial calculation and be done). – Glen_b Mar 29 '18 at 01:12
  • @Glen_b okay thank you, perhaps there was a typo in the question. Same as I have asked before, since we are looking at two different types of books in this question. Do I compute two different binomial approximations? (Because I do not know how the two approximations for each of the two books, would be able to be computed together) – queence Apr 02 '18 at 18:32
  • I'm sorry, I am unclear what you're asking there. I would be surprised if you're expected to use two different approximations; ask the people responsible for your course. [Incidentally your attempt isn't correct. Check very carefully what you're doing there] – Glen_b Apr 02 '18 at 22:38

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