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 offwithin5 minutes. For the other 500 subjects, the oil was washed offafter5 minutes. The results are summarized in the following table.
| Reaction |
Within 5 Minutes |
After 5 Minutes |
Row Total |
None
Mild
Strong |
406
65
29 |
58
345
97 |
464
410
126 |
| 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.)
| (d) |
P(N and W) |
| P(M and W) |
(e)P(N or M).
Are the eventsN= no reaction andM= mild reaction mutually exclusive? Explain.
Yes. P(N and M) = 0.No. P(N and M) ≠ 0.No. P(N or M) ≠ 0.Yes. P(N or M) = 0.
(f) Are the eventsN= no reaction andW= 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).