In: Statistics and Probability
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 |
413 57 30 |
52 322 126 |
465 379 156 |
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) | |
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.
Yes. P(N or M) = 0.No. P(N or M) ≠ 0. No. P(N and 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).
finding the probabilities
a]
b]
c]
d]
e]
for mutually exclusive events
so N = no reaction and M = mild reaction are mutually exclusive because
f] for independent events the condition is
from table we get
we get that
so they are not independent events as