Question
12) Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at the 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 other 500 subjects, the oil was washed off after 5 minutes. The results are summarized in Table 5-5. Time within 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/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.
f) Are the events N = no reaction and W- washing oil off within 5 minutes independent? Explain.
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/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.
f) Are the events N = no reaction and W- washing oil off within 5 minutes independent? Explain.
Answers
Bot
a) To find P(N), P(M), and P(S), we need to look at the row totals for each reaction and divide by the total number of subjects (1000).
P(N) = 470/1000 = 0.47
P(M) = 390/1000 = 0.39
P(S) = 140/1000 = 0.14
b) To find P(N/W) and P(S/W), we look at the probabilities within the "within 5 minutes" column.
P(N/W) = 420/500 = 0.84
P(S/W) = 20/500 = 0.04
c) Similarly, for P(N/A) and P(S/A), we look at the probabilities within the "after 5 minutes" column.
P(N/A) = 50/500 = 0.1
P(S/A) = 120/500 = 0.24
d) To find the probabilities P(N and W) and P(M and W), we look at the intersection of the respective events in the table.
P(N and W) = 420/1000 = 0.42
P(M and W) = 60/1000 = 0.06
e) To find P(N or M), we add the probabilities P(N) and P(M).
P(N or M) = P(N) + P(M) = 0.47 + 0.39 = 0.86
As there is no overlap between the events "no reaction" and "mild reaction," they are considered mutually exclusive.
f) To determine if the events N and W are independent, we need to check if P(N and W) = P(N) * P(W).
First, we need to find P(W), which is the probability of washing the oil off within 5 minutes. Since our sample is evenly split between those who washed within 5 minutes and those who didn't, P(W) = 500/1000 = 0.5.
P(N and W) = 0.42 (from part d)
P(N) * P(W) = 0.47 * 0.5 = 0.235
Since P(N and W) ≠ P(N) * P(W), the events N and W are not independent. This makes sense, as washing the oil off within 5 minutes likely affects the probability of having no reaction.
P(N) = 470/1000 = 0.47
P(M) = 390/1000 = 0.39
P(S) = 140/1000 = 0.14
b) To find P(N/W) and P(S/W), we look at the probabilities within the "within 5 minutes" column.
P(N/W) = 420/500 = 0.84
P(S/W) = 20/500 = 0.04
c) Similarly, for P(N/A) and P(S/A), we look at the probabilities within the "after 5 minutes" column.
P(N/A) = 50/500 = 0.1
P(S/A) = 120/500 = 0.24
d) To find the probabilities P(N and W) and P(M and W), we look at the intersection of the respective events in the table.
P(N and W) = 420/1000 = 0.42
P(M and W) = 60/1000 = 0.06
e) To find P(N or M), we add the probabilities P(N) and P(M).
P(N or M) = P(N) + P(M) = 0.47 + 0.39 = 0.86
As there is no overlap between the events "no reaction" and "mild reaction," they are considered mutually exclusive.
f) To determine if the events N and W are independent, we need to check if P(N and W) = P(N) * P(W).
First, we need to find P(W), which is the probability of washing the oil off within 5 minutes. Since our sample is evenly split between those who washed within 5 minutes and those who didn't, P(W) = 500/1000 = 0.5.
P(N and W) = 0.42 (from part d)
P(N) * P(W) = 0.47 * 0.5 = 0.235
Since P(N and W) ≠ P(N) * P(W), the events N and W are not independent. This makes sense, as washing the oil off within 5 minutes likely affects the probability of having no reaction.