Answer. Here p = 1 6 , so np = 30 and p np (1-p ) = 5. Then P ( S n > 50) ≈ P ( Z > 4) , which is very small. Example. Suppose a drug is supposed to be 75% eﬀective. It is tested on 100 people. What is the probability more than 70 people will be helped? Answer. Here S n is the number of successes, n = 100, and p = . 75. We have P ( S n ≥ 70) = P (( S n-75) / p 300 / 16 ≥ -1 . 154) ≈ P ( Z ≥ -1 . 154) ≈ . 87 . (The last ﬁgure came from a table.) When b-a is small, there is a correction that makes things more accurate, namely replace a by a-1 2 and b by b + 1 2 . This correction never hurts and is sometime necessary. For example, in tossing a coin 100 times, there ispositive probability that there are exactly 50 heads, while without the correction, the answer given by the normal approximation would be 0. Example. We toss a coin 100 times. What is the probability of getting 49, 50, or 51 heads? Answer.
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.