Question

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	392 54 54	46 334 120	438 388 174
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.

No. P(N or M) ≠ 0.Yes. P(N and M) = 0.    Yes. P(N or M) = 0.No. 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).

Answer 1

a)      
P(N)=438/1000=0.438      
P(M)=388/1000=0.388      
P(S)=174/1000=0.174      
      
      
      
b)      
P(N|W) = P(N n W)/P(W)=392/500=   0.784  
      
P(S|W)=P(S n W)/P(W)=54/500=   0.108  
      
      
      
c)      
P(N|A)=46/500=   0.092  
P(S|A)=120/500=   0.240  
      
      
d)      
P(N and W)=392/1000=0.392      
P(M and W)=54/1000=0.054      
      
      
e)      

Yes. because P(N and M) = 0

f)

P(N and W) = 0.392

P(N)*P(W)=0.438*0.5=0.219

answer: No. P(N and W) ≠ P(N) · P(W)