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.
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.
1 answer
1 answer