Exercise 48 If we throw a fair coin 10,000 times, what is the probability that it lands on “tail” at least 5,500 times? (i) Write down the exact probability in sum notation. (ii) Estimate the probability with the Chebyshev inequality. (iii) Estimate the probability with the Poisson‐Approximation. (iv) Estimate die probability with the normal distribution. Exercise 49 Let X ∼ NegBinom(900, 1/4 ). Estimate the probability P(X ≥ 3000)

College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.4: Expected Value
Problem 20E
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Exercise 48 If we throw a fair coin 10,000 times, what is the probability that it lands on “tail” at least 5,500 times?

(i) Write down the exact probability in sum notation.

(ii) Estimate the probability with the Chebyshev inequality.

(iii) Estimate the probability with the Poisson‐Approximation.

(iv) Estimate die probability with the normal distribution.

Exercise 49 Let X ∼ NegBinom(900, 1/4 ). Estimate the probability P(X ≥ 3000)

(i) with the Markov inequality,

(ii) with the normal distribution.

Hint: Use the fact that a negative binomial random variable can be written as a sum of geometric random variables.

Exercise 50 (i) Let Xn ∼ Poisson(10), n ≥ 1, be independent random variables. Estimate the probability that P(9 ≤ S40 ≤ 12).

(ii) Let Xn ∼ Exp(4), n ≥ 1, be independent random variables. Estimate the probability that P (1/100 * S100 ≥ 1 2).

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