Suppose you draw from a box /0/ /1/ - 400times. a.)Approximately what is the pro

Question

Suppose you draw from a box /0/  /1/ - 400times. a.)Approximately what is the probability that the sum of thedraws is between 190 and 210 (inclusive)? b.) Approximately what is the probability that the product ofthe draws is nonzero? (Hint: should you be using the Central Limit Theoremhere?) Suppose you draw from a box /0/  /1/ - 400times. a.)Approximately what is the probability that the sum of thedraws is between 190 and 210 (inclusive)? b.) Approximately what is the probability that the product ofthe draws is nonzero? (Hint: should you be using the Central Limit Theoremhere?)

Explanation / Answer

My assumption (I'm not totally clear on the problem)... Binomial distribution p = .5, n = 400. Success is a draw of1, failure is a draw of 0. To apply the Central Limit Theorem, we need np > 5 and n(1-p)> 5. 400 * .5 = 200 400 * (1-.5) = 200 Thus, this distribution can be approximated by a normal N(np, sqrt(n*p*(1-p)). np = 200 sqrt(n*p*(1-p)) = 10 Thus, ignoring the discontinuity correction (doing so cause it's alarge distribution and will have a negligible effect), we arelooking at z = (190 - 200) / 10 = 1 to z = (210 - 200)/10 = 1. Going from -1 to 1 stdev encompasses 68% of the distribution. Looking for a non-zero product, we can't use the normaldistribution because it is continuous, and the probability of acontinuous distribution equaling anything is 0. Thus, we use the exact answer of P(product is non-zero) = P(all1's) = .5^400 = 1/2^400 = 3.873 ^ (-121).

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