Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at the 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 Table 5-5.
Time Within Which Oil Was Washed Off

Let€™s use the following notation for the various events: W = washing oil off within 5 minutes, A ˆ’ washing oil off after 5 minutes, N ˆ’ no reaction, M ˆ’ mild reaction, S = strong reaction. Find the following probabilities for a person selected at random from this sample of 1000 subjects.
(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 ˆ’ no reaction and M ˆ’ mild reaction mutually exclusive? Explain.
(f) Are the events N = no reaction and W = washing oil off within = minutes independent? Explain.

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Reaction None Mild Strong Column Total Within 5 Minutes 420 60 20 500 After 5 Minutes 50 330 120 500 Row Total 470 390 140 1000