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

Reaction	Within 5 Minutes	After 5 Minutes	Row Total
None Mild Strong	392 54 54	46 334 120	438 388 174
Column Total	500	500	1000

Let's use the following notation for the various events:W = washing oil off within 5 minutes, A = washingoil off after 5 minutes, N = no reaction, M =mild reaction, S = strong reaction. Find the followingprobabilities for a person selected at random from this sample of1000 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 = mildreaction mutually exclusive? Explain.

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

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

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