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.
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.
a) P(N) = (Number of people with no reaction)/(Total number of people) = 470/1000 = 0.47
P(M) = (Number of people with mild reaction)/(Total number of people) = 390/1000 = 0.39
P(S) = (Number of people with strong reaction)/(Total number of people) = 140/1000 = 0.14
b) P(N/W) = (Number of people with no reaction and oil washed off within 5 minutes)/(Total number of people with oil washed off within 5 minutes) = 420/500 = 0.84
P(S/W) = (Number of people with strong reaction and oil washed off within 5 minutes)/(Total number of people with oil washed off within 5 minutes) = 20/500 = 0.04
c) P(N/A) = (Number of people with no reaction and oil washed off after 5 minutes)/(Total number of people with oil washed off after 5 minutes) = 50/500 = 0.1
P(S/A) = (Number of people with strong reaction and oil washed off after 5 minutes)/(Total number of people with oil washed off after 5 minutes) = 120/500 = 0.24
d) P(N and W) = (Number of people with no reaction and oil washed off within 5 minutes)/(Total number of people) = 420/1000 = 0.42
P(M and W) = (Number of people with mild reaction and oil washed off within 5 minutes)/(Total number of people) = 330/1000 = 0.33
e) P(N or M) = P(N) + P(M) - P(N and M)
= 0.47 + 0.39 - (Number of people with no reaction and mild reaction)/(Total number of people)
= 0.47 + 0.39 - 0
= 0.86
The events N = no reaction and M = mild reaction are not mutually exclusive since there can be people who have both no reaction and mild reaction.
f) The events N = no reaction and W = washing oil off within 5 minutes are independent if P(N and W) = P(N) * P(W).
P(N and W) = 0.42
P(N) = 0.47
P(W) = (Total number of people with oil washed off within 5 minutes)/(Total number of people)
= 500/1000 = 0.5
P(N) * P(W) = 0.47 * 0.5 = 0.235
Since P(N and W) = 0.42 and P(N) * P(W) = 0.235, the events N = no reaction and W = washing oil off within 5 minutes are not independent.
To find the probabilities for the given events, we will use the data provided in Table 5-5.
a) P(N), P(M), P(S):
- P(N) is the probability of having no reaction. It is given by the count of subjects with no reaction divided by the total number of subjects: P(N) = 470/1000 = 0.47
- P(M) is the probability of having a mild reaction. It is given by the count of subjects with a mild reaction divided by the total number of subjects: P(M) = 390/1000 = 0.39
- P(S) is the probability of having a strong reaction. It is given by the count of subjects with a strong reaction divided by the total number of subjects: P(S) = 140/1000 = 0.14
b) P(N/W), P(S/W):
- P(N/W) is the probability of having no reaction given that the oil was washed off within 5 minutes. It is given by the count of subjects with no reaction and the oil washed off within 5 minutes divided by the total number of subjects with the oil washed off within 5 minutes: P(N/W) = 420/500 = 0.84
- P(S/W) is the probability of having a strong reaction given that the oil was washed off within 5 minutes. It is given by the count of subjects with a strong reaction and the oil washed off within 5 minutes divided by the total number of subjects with the oil washed off within 5 minutes: P(S/W) = 20/500 = 0.04
c) P(N/A), P(S/A):
- P(N/A) is the probability of having no reaction given that the oil was washed off after 5 minutes. It is given by the count of subjects with no reaction and the oil washed off after 5 minutes divided by the total number of subjects with the oil washed off after 5 minutes: P(N/A) = 50/500 = 0.1
- P(S/A) is the probability of having a strong reaction given that the oil was washed off after 5 minutes. It is given by the count of subjects with a strong reaction and the oil washed off after 5 minutes divided by the total number of subjects with the oil washed off after 5 minutes: P(S/A) = 120/500 = 0.24
d) P(N and W), P(M and W):
- P(N and W) is the probability of having no reaction and the oil washed off within 5 minutes. It is given by the count of subjects with no reaction and the oil washed off within 5 minutes divided by the total number of subjects: P(N and W) = 420/1000 = 0.42
- P(M and W) is the probability of having a mild reaction and the oil washed off within 5 minutes. It is given by the count of subjects with a mild reaction and the oil washed off within 5 minutes divided by the total number of subjects: P(M and W) = 60/1000 = 0.06
e) P (N or M):
- P (N or M) is the probability of having either no reaction or a mild reaction. It is given by the sum of the probabilities of P(N) and P(M): P(N or M) = P(N) + P(M) = 0.47 + 0.39 = 0.86
We can see that these events are not mutually exclusive because there are some subjects who have both no reaction and a mild reaction.
f) Are the events N = no reaction and W- washing oil off within 5 minutes independent?
To determine if events N and W are independent, we need to compare their joint probability, P(N and W), with the product of their individual probabilities, P(N) and P(W):
- P(N and W) = 0.42 (from part d)
- P(N) = 0.47 (from part a)
- P(W) is the probability of the oil being washed off within 5 minutes, which is given in Table 5-5 as 500/1000 = 0.5
If P(N and W) = P(N) * P(W), then events N and W are independent.
0.42 is not equal to 0.47 * 0.5, so events N and W are not independent.