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Exercise 4.41 (p. 153 of the text book) An experiment is conducted to test the effect of an anticoagulant drug on rats. A random sample of 4 rats is employed in the experiment. If the drug manufacturer's claim that 80% of rats given the drug will be favorably affected by the drug is indeed true, answer the following questions. First, define the Binomial random variable X as the number of rats favorably affected out of a random sample of 4 rats. Then X is distributed as the Binomial R.V. with the distribution Bin(4,0.8) (i) What is the probability that none of the 4 rats will be favorably affected? ⎛4⎞ P(X=0)= ⎜⎜ ⎟⎟(0.8) 0 (1 − 0.8) 4 = 1 × 1 × 0.0016 = 0.0016 ⎝0⎠ (ii) What is the probability that one of the 4 rats will be favorably affected? ⎛4⎞ P(X=1)= ⎜⎜ ⎟⎟(0.8)1 (1 − 0.8) 3 = 4 × 0.8 × 0.008 = 0.0256 ⎝1⎠ (iii) What is the probability that one or fewer of the 4 rats will be favorably affected? P(X ≤ 1) = P(X=0 or X=1) = P(X=0)+ P(X=1) = 0.0016 + 0.0256 = 0.0272 (iv) How many of 100 rats given the drug would you expect to be favorably affected? What property of X are you using ? In this case, X is distributed as Bin(100,0.8) Expected value of X = E(X) = nπ = 100 × 0.8 = 80