Reaction to Poison Ivy Allergic reactions to poison ivy can be miserable. Plant

Question

Reaction to Poison Ivy
Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at Allergy Institute did a study to see 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 table:

Time in Which Oil Was Washed Off

Reaction Within 5 Minutes After 5 Minutes Row Total
None 420 50 470
Mild 60 330 390
Strong 20 120 140
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.
(a.) P(N), P(M), P(S)
(b.) P(N, given W), P(S, given W)
(c.) P(N, given A), P(S, given 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.
(f.) Are the events N = no reaction and W = washing oil off within 5 minutes independent? Explain.

Explanation / Answer

a) P(N)=470/1000=0.47;P(M)=390/1000;P(S)=140/1000=0.14

b)P(N/W)=420/500=0.84;P(S/W)=20/500=0.04

c)P(N/A)=50/500=0.10;P(S/A)=120/500=0.24

d)P(N and W)=420/1000=0.42 ; P(M AND W)=60/1000=0.6

e)P(N or M)=0.47+0.39=0.86;P(N and M)=P(N)*P(M) . They are mutually exclusive because They've nothing in common

f)P(no reaction/within 5 min)=P(no reaction) 0.84!=0.47 . So they are not independent

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