3. [-/12 Points] MY NOTES DETAILS BBUNDERSTAT ASK YOUR TEACHER 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 1,000 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. Time Within Which Oil Was Washed Off Reaction None Within 5 Minutes After 5 Minutes Row Total 399 70 469 Mild 66 342 408 Strong Column Total 35 88 123 500 500 1,000 washing Let's use the following notation for the various events: W = washing oil off within 5 minutes, A = oil off after 5 minutes, N = no reaction, M = mild reaction, S = strong reaction. Suppose a person is selected at random from this sample of 1,000 subjects. (a) Find the following probabilities. (i) P(N) (ii) P(M) (iii) P(S) (b) Find the following probabilities. (i) P(NW) (ii) P(S| W) (c) Find the following probabilities. (i) P(NA) (ii) P(S|A) (d) Find the following probabilities. (i) P(N and W) (ii) P(M and W) (e) Calculate the probability P(N or M). Are the events N = no reaction and M = mild reaction mutually exclusive? Explain. No. P(N and M) # 0. No. P(N or M) # 0. Yes. P(N or M) = 0. Yes. 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). O No. P(N and W) = P(N) P(W). O No. P(N and W) + P(N) P(W). Yes. P(N and W) = P(N) P(W).