Question

Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at Allergy Institute did a study to determine the effects of washing the oil off within 5 minutes of exposure. A random sample of 1000 people with known allergies to poison ivy participated in the study. Oil from the poison ivy plant was rubbed on a patch of skin. For 500 of the subjects, it was washed off within 5 minutes. For the other 500 subjects, the oil was washed off after 5 minutes. The results are summarized in the following table.

Reaction	Within 5 Minutes	After 5 Minutes	Row Total
None Mild Strong	408 52 40	58 336 106	466 388 146
Column Total	500	500	1000

Let's use the following notation for the various events: W = washing oil off within 5 minutes, A = washing oil off after 5 minutes, N = no reaction, M = mild reaction, S = strong reaction. Find the following probabilities for a person selected at random from this sample of 1000 subjects. (Use 3 decimal places.)

(a)	P(N)
	P(M)
	P(S)

(b)	P(N | W)
	P(S | W)

(c)	P(N | A)
	P(S | A)

(d)	P(N and W)
	P(M and W)

(e) P(N or M).

Are the events N = no reaction and M = mild reaction mutually exclusive? Explain.

Yes. P(N and M) = 0.Yes. P(N or M) = 0.     No. P(N and M) ≠ 0.No. P(N or M) ≠ 0.

(f) Are the events N = no reaction and W = washing oil off within 5 minutes independent? Explain.

No. P(N and W) ≠ P(N) · P(W).Yes. P(N and W) ≠ P(N) · P(W).     Yes. P(N and W) = P(N) · P(W).No. P(N and W) = P(N) · P(W).

Answer 1

(a)

(b)

For the 500 subjects for which events "washing oil off within 5 minutes" occur, 408 shows no reaction so

P(N | W) = 408 /500 = 0.816

Likewise

P(S|W) = 40 /500 = 0.080

(c)

P(N|A) = 58 /500 =0.116

P(S|A) = 106 / 500= 0.212

(d)

P(N and W) = 408 /1000 = 0.408

P(M and W) = 52 /1000 = 0.052

(e)

P(N or M) = P(N) + P(M) = 0.466+ 0.388 = 0.854

Since P(N and M) = 0 so the events N = no reaction and M = mild reaction are mutually exclusive.

(f)

P(W) = 500/1000 = 0.50

P(N) * P(W) = 0.466 *0.50 = 0.233

Correct option:

No. P(N and W) ≠ P(N) · P(W).