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 | Row Total |
None Mild Strong |
407 51 42 |
54 332 114 |
461 383 156 |
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) | 0.461 |
P(M) | 0.383 | |
P(S) | 0.156 |
(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).
0.844
Are the events N = no reaction and M = mild
reaction mutually exclusive? Explain.
Yes. P(N or M) = 0.
No. P(N and M) ≠ 0.
No. 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).
Yes. P(N and W) ≠ P(N) · P(W).
No. P(N and W) = P(N) · P(W).
No. P(N and W) ≠ P(N) · P(W).
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