Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction

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.

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.)

(e) P(N or M).

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

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

Reaction Within 5 Minutes After 5 Minutes Row Total None
Mild
Strong 404
69
27 41
340
119 445
409
146 Column Total 500 500 1000

Explanation / Answer

a) P(N) = 445/1000 = 0.445

P(M) =409/1000 = 0.409

P(S) = 146/1000 = 0.146

b) P(N|W) =P(N and W)/P(W) = 404/500 =0.808

P(S|W) =P(Sand W)/P(W) = 27/500 = 0.054

c) P(N|A) = P(N and A)/P(A) = 41/500 =0.082

P(S|A) =P(S and A) /P(A) = 119/500 = 0.238

d) P(N and W) = 404/1000 = 0.404

P(M and W) = 69/1000 = 0.069

e) P(N or M) = P(N)+P(M) -P(N and M) = 0.445+ 0.409 -0 = 00.854

Yes P(N and M) = 0

f) P(Nand W) = 0.404

P(N)*P(W) = 0.445*0.5 =0.2225

Answer: No. P(N and W) P(N) · P(W)

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