5. A fair die is rolled 720 times. Let S be the event that there are at least 130 sixes rolled and F be the event that exactly 100 fives are rolled. Estimate P(S) and P(S | F) using the de Moivre-Laplace Central Limit Theorem (which we will cover Friday). Leave your answers in terms of the function Φ.
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Then $X$ is a binomial random variable with parameters $n = 720$ and $p = 1/6$. Show more…
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Madhur L.
(4). A fair six-sided die is repeatedly rolled until the total sum of all rolls exceeds 300. Use the Central Limit Theorem to approximate the probability that at least 80 rolls are necessary. (Hint: Let Xi be the outcome of the ith roll. You can use the fact that X1, X2, ... are iid random variables with E[Xi] = 7/2 and Var(Xi) = 35/12. Consider X = sum_{i=1}^{79} Xi. Then P(X <= 300) is the same as the probability that at least 80 rolls are necessary.)
Lucas F.
A fair six-sided die is repeatedly rolled until the total sum of all rolls exceeds 300. Use the Central Limit Theorem to approximate the probability that at least 80 rolls are necessary. (Hint: Let Xi be the outcome of the ith roll. You can use the fact that X1, X2,... are iid random variables with E[Xi] = 7/2 and Var(Xi) = 35/12. Consider X = sum_{i=1}^{79} Xi. Then P(X <= 300) is the same as the probability that at least 80 rolls are necessary.)
David N.
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