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	396	50	446
Mild	61	327	388
Strong	43	123	166
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.)

1. P(N)=

P(M)=

P(S)=

2. P(N | W)=

P(S | W)=

3. P(N | A)=

P(S | A)=

4. P(N and W)=

P(M and W)=

P(N or M)=

Answer 1

Solution(1a)
We need to calculate P(N) = ?
Here N is No reaction
P(N). can be calculated as
P(N) = Total number of No reaction / Total No. of sample = 446/1000 = 0.446
So there is 44.6% probability that that is no reaction
Solution(1b)
We need to calculate P(M) = ?
Here M is Mild reaction
P(M) can be calculated as
P(M) = Total number of Mild reaction / Total No. of sample = 388/1000 = 0.388
So there is 38.8% probability that that is mild reaction
Solution(1c)
We need to calculate P(S) = ?
Here S is strong reaction
P(S) can be calculated as
P(S) = Total number of Strong reaction / Total No. of sample = 166/1000 = 0.166
So there is 16.6% probability that that is strong reaction.
Solution(2a)
We need to calculate P(N|W) which can be calculated as
P(N | W) = P(No reaction and within 5 miutes)/P(W) = (396/1000)/(500/1000) = 0.792
P(N | W) = 0.792 or 79.2%
Solution(2b)
We need to calculate P(S | W) which can be calculated as
P(S | W) = P(Strong reaction and within 5 miutes)/P(W) = (43/1000)/(500/1000) = 0.086
P(N | W) = 0.0.086 or 8.6%
Solution(3a)
We need to calculate P(N | A) which can be calculated as
P(N | A) = P(No reaction and After 5 miutes)/P(A) = (50/1000)/(500/1000) = 50/500 = 0.1
P(N | A) = 0.1 or 10%
Solution(3b)
We need to calculate P(S | A) which can be calculated as
P(S | A) = P(Strong reaction and After 5 miutes)/P(A) = (123/1000)/(500/1000) = 0.246
P(S | A) = 0.246
Solution(4a)
We need to calculate P(N and W) which can be calculated as
P(N and W) = P(No reaction and within 5 miutes) = (396/1000) = 0.396
P(N and W) = 0.396
Solution(4b)
We need to calculate P(M and W) which can be calculated as
P(M and W) = P(Mild reaction and within 5 miutes) = (61/1000) = 0.061
P(M and W) = 0.061
Solution(4c)
We need to calculate P(N or M) which can be calculated as
P(N or M) = P(No reaction) + P(Mild Reaction) - P(No reaction and Mild reaction) = 446/1000 + 388/1000 -0/1000 = 834/1000 = 0.834
P(N or M) = 0.834