STA104/MTH135: Probability

Homework #3: Binomial & Normal

Text Problems are from Jim Pitman, Probability (1st edn).

• Exercises
  

  § 2.1:		1	6	10	14
  § 2.2:		2	4	9	12
  § 2.3:		1	2

• Another Problem (Optional, extra credit)
  We suspect that our die might be biased- and so the probaility of an ace might not be exactly 1/6. Let's count the number X of times an Ace appears in N independent rolls, and estimate the probabliity p of an ace as the frequency X/N of aces in the first N rolls (an "Ace" is just a roll of 1).
  
  How big must N be to ensure that the probability P[ |p - X/N| < 0.01] is at least 0.99--- i.e., how large a sample do we need to ensure that the estimate X/N of the probability p of an ace for a possibly-unfair die will be within ±0.01 of the correct answer, with probability at least 0.99 ?
  
  It's easy to show that N=16587 is big enough by using a normal approximation, being careless about the ±½ terms, and maximizing over all 0≤p≤1. See if you can do better--- either by being careful about the ±½ terms, or using MatLab or R to compute the binomial distribution probabilities without using a normal approximation. Show your work. Would the answer change if you are allowed to assume that p &le 0.20?

Good Luck!