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