a)
P(P) = P(P|S)*P(S) + P(P|N)*P(N)
= 0.9 * 0.3 + 0.1 * 0.7
= 0.34
b)
P(P and S) = P(P|S) * P(S)
= 0.9 * 0.3
= 0.27
c)
P(S|P) = P(P|S)*P(S)/(P(P|S)*P(S) + P(P|N)*P(N))
= 0.9 * 0.3 / (0.9 * 0.3 + 0.1 * 0.7)
= 0.7941
5. Consider a genetic test for susceptibility to a certain environmentally induced illness. Let S denote...
Consider randomly selecting a student at a certain
university, and let ? denote the event that the selected individual
has a Visa credit card and ? the analogous event for a MasterCard.
Suppose that ?(?) = 0.5, ?(?) = 0.4, and ?(? ∪ ?) = 0.65.
a. What is the probability that the student has both types of
cards?
b. What is the probability that the student has a MasterCard but
not a Visa?
c. What is the probability the...
Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for MasterCard. Suppose that P(A) 0.5, P(B) 0.4, and P(An B) 0.25. Calculate and interpret each of the following probabilities. b. P BIA) f. Is having a Visa credit card and a MasterCard independent? Justify your answer
4. For a diagnostic test of a certain disease, let T1 denote the probability that the diagnosis is positive given that a subject has the disease, and let T2 denote the probability that the diagnosis is positive given that a subject does not have it. Let p denote the probability that a subject has the disease. (a) More relevant to a patient who has received a positive diagnosis is the probability that they truly have the disease. Given that a...
Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard. Suppose that P(A) = 0.3, P(B) = 0.4, and P(A ∩ B) = 0.05.(a) Compute the probability that the selected individual has at least one of the two types of cards (i.e., the probability of the event A ∪B).(b) What is the probability that the selected individual has neither type of card?(c) Describe, in terms of A and B, the event that the selected...
Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard where P(A) = 0.45, P(B) = 0.35, and P(A ❩ B) = 0.30. Calculate and interpret each of the following probabilities (a Venn diagram might help). (Round your answers to four decimal places.) (a) P(B | A) (b) P(B' | A) (c) P(A | B) (d) P(A' | B) (e) Given...
3. Let C be the event that a patient suffers from a certain condition, and let T denote a positive result from a lab test that is designed to detect the presence of said condi- tion. Suppose that the proportion of the population that actually has the condition IS E E (0,1). Additionally, suppose that, when the condition is actually present in a patient, the test is positive with probability a (0,1). On the other hand, when the patient does...
9) Suppose that a laboratory test to detect a certain disease has the following statistics. Let A- event that the tested person has the disease B-event that the test result is positive It is known that P(BIA) 0.99 and P(BIA) 0.005 and 0.1% of the population actually has the disease, what is the probability that a person has the disease given that the test result is positive?
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
Problem #7: A certain system can experience three different types of defects. Let Ai (1 1,2,3) denote the event that the system has a defect of type i. Suppose that P(41) = 0.3 l, P(d2) = 0.29, Padg) = 0.35, P(A2 U A3) = 0.59. P(41 n.42 n A3) = 0.03 (a) Find the probability that the system has exactly 2 of the 3 types of defects. (b) Find the probability that the system has a type 1 defect given...
Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing Ho: p=0.5 versus Ha: p?0.5 based on a random sample of 25 individuals. Let the test statistic X be the number in the sample who favor the first company and x represent the observed value of X. a) Describe type I and type II errors in the...