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).