Suppose you roll a fair, 4-sided die once every day for 360 days.[Image source: https://commons.wikimedia.org/wiki/File:4-sided_dice_250.jpg (Links to an external site.) ; Author: Fantasy; CC BY-SA 3.0 (Links to an external site.)]Let X represent the number of rolls that land with "1" at the bottom edge of the die, as in the illustration above. Note: the probability of this event is 0.25 in any single roll.X is a binomial random variable. What are the parameters n and p for its distribution?Without evaluating the expression, use the binomial formula to write out exactly P(X = 81). You should not evaluate the binomial coefficient, nor the exponential expressions. Do not give a decimal approximation; use Formula 5.1 from our textbook (Section 5.3).We want to use the normal approximation to the binomial to approximate the probability P(85 < X ≤ 95).Check that normal approximation is reasonable in this case. (This is Step 2 of Procedure 6.3, p. 299 in our textbook.)What are the mean and standard deviation of the appropriate normal distribution? (Step 3 of Procedure 6.3.)What is the correction for continuity to evaluate this probability? I.e., what are the endpoints a and b so that, if Y denotes the normal random variable, we have P(85 < X ≤ 95) ≈ P(a ≤ Y ≤ b) ?Find the probability given by the approximation. If possible, also find the exact value of P(85 < X ≤ 95), using technology. How does the approximation compare to the exact value? 4

Given information: A fair 4-sided die is rolled once every day for 360 days. X = number of rolls…

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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Suppose you roll a fair, 4-sided die once every day for 360 days.

[Image source: https://commons.wikimedia.org/wiki/File:4-sided_dice_250.jpg (Links to an external site.) ; Author: Fantasy; CC BY-SA 3.0 (Links to an external site.)]

Let X represent the number of rolls that land with "1" at the bottom edge of the die, as in the illustration above. Note: the probability of this event is 0.25 in any single roll.

X is a binomial random variable. What are the parameters n and p for its distribution?
Without evaluating the expression, use the binomial formula to write out exactly P(X = 81). You should not evaluate the binomial coefficient, nor the exponential expressions. Do not give a decimal approximation; use Formula 5.1 from our textbook (Section 5.3).
We want to use the normal approximation to the binomial to approximate the probability P(85 < X ≤ 95).
1. Check that normal approximation is reasonable in this case. (This is Step 2 of Procedure 6.3, p. 299 in our textbook.)
2. What are the mean and standard deviation of the appropriate normal distribution? (Step 3 of Procedure 6.3.)
3. What is the correction for continuity to evaluate this probability? I.e., what are the endpoints a and b so that, if Y denotes the normal random variable, we have
  P(85 < X ≤ 95) ≈ P(a ≤ Y ≤ b) ?
4. Find the probability given by the approximation. If possible, also find the exact value of P(85 < X ≤ 95), using technology. How does the approximation compare to the exact value?

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