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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 Mild Strong 405 61 34 69 325 106 474 386 140 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. (Use 3 decimal places.)
Sri K.
Ivy League Admissions. High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2,851 applications for early admission. Of this group, it admitted 1,033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted $18 \%$ of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was $2,375 .$ Let $E, R,$ and $D$ represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate $P(E), P(R),$ and $P(D)$ b. Are events $E$ and $D$ mutually exclusive? Find $P(E \cap D)$ c. For the 2,375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission? d. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?
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 Mild Strong 419 57 24 51 322 127 470 379 151 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.) (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. Yes. P(N and M) = 0. No. P(N or M) ≠ 0. No. P(N and M) ≠ 0. Yes. P(N or M) = 0. (f) Are the events N = no reaction and W = washing oil off within 5 minutes independent? Explain. Yes. P(N and W) = P(N) · P(W). No. P(N and W) ≠ P(N) · P(W). Yes. P(N and W) ≠ P(N) · P(W). No. P(N and W) = P(N) · P(W).
Adi S.
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