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) P(N) = (number of subjects with no reaction)/(total number of subjects) = 470/1000 = 0.47

P(M) = (number of subjects with mild reaction)/(total number of subjects) = 390/1000 = 0.39
P(S) = (number of subjects with strong reaction)/(total number of subjects) = 140/1000 = 0.14

b) P(N/W) = (number of subjects with no reaction and oil washed off within 5 minutes)/(total number of subjects with oil washed off within 5 minutes) = 420/500 = 0.84
P(S/W) = (number of subjects with strong reaction and oil washed off within 5 minutes)/(total number of subjects with oil washed off within 5 minutes) = 20/500 = 0.04

c) P(N/A) = (number of subjects with no reaction and oil washed off after 5 minutes)/(total number of subjects with oil washed off after 5 minutes) = 50/500 = 0.1
P(S/A) = (number of subjects with strong reaction and oil washed off after 5 minutes)/(total number of subjects with oil washed off after 5 minutes) = 120/500 = 0.24

d) P(N and W) = (number of subjects with no reaction and oil washed off within 5 minutes)/(total number of subjects) = 420/1000 = 0.42
P(M and W) = (number of subjects with mild reaction and oil washed off within 5 minutes)/(total number of subjects) = 60/1000 = 0.06

e) P(N or M) = P(N) + P(M) = 0.47 + 0.39 = 0.86
The events N = no reaction and M = mild reaction are not mutually exclusive, as there are subjects who had both no reaction and a mild reaction.

f) The events N = no reaction and W = washing oil off within 5 minutes are not independent because P(N and W) is not equal to P(N) * P(W). If the events were independent, the probability of having no reaction and washing off the oil within 5 minutes would be the product of the individual probabilities, which is not the case here.

To find the probabilities, we will use the information provided in the table. Let's calculate them step by step:

a) To find P(N), P(M), P(S), we will divide the count of each reaction by the total number of subjects.

P(N) = Number of subjects with no reaction / Total number of subjects = 470 / 1000 = 0.47
P(M) = Number of subjects with mild reaction / Total number of subjects = 390 / 1000 = 0.39
P(S) = Number of subjects with strong reaction / Total number of subjects = 140 / 1000 = 0.14

b) To find P(N/W), P(S/W), we will divide the count of each reaction within 5 minutes by the total number of subjects who washed off the oil within 5 minutes.

P(N/W) = Number of subjects with no reaction and washed off within 5 minutes / Total number of subjects who washed off within 5 minutes = 420 / 500 = 0.84
P(S/W) = Number of subjects with strong reaction and washed off within 5 minutes / Total number of subjects who washed off within 5 minutes = 20 / 500 = 0.04

c) To find P(N/A), P(S/A), we will divide the count of each reaction after 5 minutes by the total number of subjects who washed off the oil after 5 minutes.

P(N/A) = Number of subjects with no reaction and washed off after 5 minutes / Total number of subjects who washed off after 5 minutes = 50 / 500 = 0.10
P(S/A) = Number of subjects with strong reaction and washed off after 5 minutes / Total number of subjects who washed off after 5 minutes = 120 / 500 = 0.24

d) To find P(N and W), P(M and W), we will divide the count of each specific combination by the total number of subjects.

P(N and W) = Number of subjects with no reaction and washed off within 5 minutes / Total number of subjects = 420 / 1000 = 0.42
P(M and W) = Number of subjects with mild reaction and washed off within 5 minutes / Total number of subjects = 60 / 1000 = 0.06

e) To find P(N or M), we will add the probabilities of each individual event, as long as they are not mutually exclusive. Two events are mutually exclusive if they cannot happen at the same time (if the occurrence of one event excludes the occurrence of the other).

P(N or M) = P(N) + P(M) = 0.47 + 0.39 = 0.86

To check if N and M are mutually exclusive, we need to ensure that the intersection between them is empty. In this case:

P(N and M) = Number of subjects with no reaction and mild reaction / Total number of subjects = 0
Since the intersection is empty, N and M are mutually exclusive.

f) To check if N and W are independent, we need to compare P(N and W) with P(N) * P(W). If these two probabilities are equal, then the events are independent.

P(N and W) = Number of subjects with no reaction and washed off within 5 minutes / Total number of subjects = 0.42
P(N) = Number of subjects with no reaction / Total number of subjects = 0.47
P(W) = Total number of subjects who washed off within 5 minutes / Total number of subjects = 0.5

P(N) * P(W) = 0.47 * 0.5 = 0.235

Since P(N and W) = P(N) * P(W), we can conclude that N and W are independent events.

Note: Please ensure you have checked the table correctly for all the counts mentioned in the calculations.

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