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 RowTotal

None 418 69 487

Mild 56 328 384

Strong 26 103 129

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.

No. P(N and M) ≠ 0.

Yes. P(N or M) = 0.    

Yes. 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).

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).