1 Allergy sufferers often take Loratadine or Cetirizine Supp

1. Allergy sufferers often take Loratadine or Cetirizine. Suppose 70% of allergy sufferers have taken Loratadine at some point in their life, while 79% have taken Cetirizine. Furthermore, 83% have had either Loratadine OR Cetirizine. Given that a randomly selected person has taken Loratadine at some point in their life, what is the probability that this person has also taken Cetirizine? (Hint: Draw a Venn Diagram)

2. Freckles and moles on humans are very common. However usually large, abnormally shaped and/or abnormally colored freckles/moles can sometimes be warning signs of skin cancer.
Let X be the number of “unusual” freckles/moles found on patients that are seen by a dermatologist’s office in Bloomington. We’ll assume that an individual can only have between 0 and 4 unusual moles/freckles each, hence X = {0,.....,4}. The population probability distribution for the number of unusual freckles/moles is presented in the following table.

0.05

(a) Find P(X 2).
(b) Find P(X > 1).
(c) Find P[X < 2 or X 3].
(d) Find X.
(e) Find (2/X)

3. If X and Y are independent random variables with means X = 9.5 and Y = 6.8 and standard deviations X = 0.4 and Y = 0.1, find the means and standard deviations of the following:

(a) W = 3Y - 2X, hence find W and W.
(b) Z = X +4Y - 0.2, hence find Z and Z.

4. At a large university, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn. What is the probability that the average age of these students is greater than 23 years? (Hint: The sample size is large and you know , so you can use the central limit theorem.)

Solution

1.

Let

L = taken Loratadine
C = taken Cetirizine

Thus,

P(C|L) = P(C n L) / P(L)

As

P(C n L) = P(C) + P(L) - P(C U L) = 0.70+0.79-0.83 = 0.66

Then

P(C|L) = 0.66 / 0.70 = 0.942857143 [answer]

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