Unformatted text preview: Chapter 4: Condi/onal Probability and Independence h6p://imgs.xkcd.com/comics/condi/onal_risk.png Example: Condi/onal Probability Roll a fair 4 sided die 3 /mes A = the event that two 1’s are tossed B = the event that the ﬁrst roll is an 1 C = the event that the second roll is an 1 Find: P(B), P(BA), P(BC) Example: Condi/onal Probability A bus arrives punctually at a bus stop every half hour. Each morning, a commuter named Sarah leaves her house and casually strolls to the bus stop. Find the probability that that wait /me is at least 10 minutes. Find the probability that the wait /me is at least 10 minutes given that if someone is wai/ng there for more than 15 minutes, they will get a ride from a passing car. Example: Condi/onal Probability Assume the faculty member is selected at random. a) What is the probability that the faculty member is a full professor? b) What is the probability that the faculty member is a full professor given that we know that the person is in their 30’s? Table 4.1(2.5): Age and rank of faculty members Rank
Full
Associate
Professor Professor
R1
R2 Assistant
Professor
R3 Instructor
R4 Total 2 3 57 6 68 30  39
A2
Age Under 30
A1 52 170 163 17 402 40  49
A3 156 125 61 6 348 50  59
A4 145 68 36 4 253 60 & older
A5 75 15 3 0 93 Total 430 381 320 33 1164 Table 4.2: Condi/onal distribu/ons of rank by age Rank
Full
Associate
Professor Professor
R1
R2 Assistant
Professor
R3 Instructor
R4 Total 0.029 0.044 0.838 0.088 1.000 30  39
A2
Age Under 30
A1 0.129 0.423 0.405 0.042 1.000 40  49
A3 0.448 0.359 0.175 0.017 1.000 50  59
A4 0.573 0.269 0.142 0.016 1.000 60 & older
A5 0.806 0.161 0.032 0.000 1.000 P(Rj) 0.369 0.327 0.275 0.028 1.000 Example: Condi/onal Probability Rule Roll a fair 4 sided die 3 /mes A = the event that two 1’s are tossed B = the event that the ﬁrst roll is an 1 C = the event that the second roll is an 1 Find: P(BA), P(BC) Example: Condi/onal Probability Rule If the faculty member is selected at random Using the joint probability distribu/on, what is the probability that the faculty member is a full professor given that we know that the person is in their 30’s? Table 2.6: Joint probability distribu/on corresponding to Table 2.5 Rank
Full
Associate
Professor Professor
R1
R2 Assistant
Professor
R3 Instructor
R4 P(Aj) 0.002 0.003 0.049 0.005 0.058 30  39
A2
Age Under 30
A1 0.045 0.146 0.140 0.015 0.345 40  49
A3 0.134 0.107 0.052 0.005 0.299 50  59
A4 0.125 0.058 0.031 0.003 0.217 60 & older
A5 0.064 0.013 0.003 0.000 0.080 P(Rj) 0.369 0.327 0.275 0.028 1.000 Example: Condi/onal Probability Rule From the set of all families with two children, a family is selected at random and is found to have a girl. What is the probability that the other child of the family is a girl? From the set of all families with two children, a child is selected at random and is found to be a girl. What is the probability that the other child of the family is a girl? Example: General Mul/plica/on Law A consul/ng ﬁrm is awarded 51% of the contracts it bids on. Suppose that Melissa works for a division of the ﬁrm that gets to do 25% of the projects contracted for. If Melissa directs 41% of the projects submi6ed to her division, what percentage of all bids submi6ed by the ﬁrm will result in contracts for projects directed by Melissa? Example: General Mul/plica/on Law Supposed that 8 good and 2 defec/ve fuses have been mixed up. To ﬁnd the defec/ve fuses we need to test them one by one, at random. Once we test a fuse, we set it aside. What is the probability that we ﬁnd both of the defec/ve fuses in exactly three tests? Example: The Law of Total Probability John ﬂies frequently and likes to upgrade his seat to ﬁrst class. He has determined that if he checks in for his ﬂight at least 2 hours early, the probability that he will get the upgrade is 0.8; otherwise, the probability that he will get the upgrade is 0.3. With his busy schedule, he checks in at least 2 hours before his ﬂight only 40% of the /me. What is the probability that for a randomly selected trip John will be able to upgrade to ﬁrst class? Example: Independence Roll a red 4 sided die and a white 4 sided die. Let A: event that the red die is a 1 B: event that the white die is a 1 C: event that the sum of the two dice is 4 a) Are events A and B independent? b) Are events A and C independent? Example: Pairwise Independence Roll a red 4 sided die and a white 4 sided die. Let A: event that the red die is even B: event that the white die is even C: event that the sum of the two dice is even a) Show that A, B, and C are pairwise independent. b) Show that A ∩ B and C are NOT independent. Example: Mutual Independence Roll a red 6 sided die and a white 6 sided die. Let A: event that the red die is 1 or 2 or 3 B: event that the white die 4 or 5 or 6 C: event that the sum of the two dice is 5 Show that P(D ∩ E ∩ F) = P(D)P(E)P(F) but D, E and F are NOT (mutually) independent events. Example: Electrical Components The ﬁgure below shows an electric circuit in which each of the switches located at 1,2,3, and 4 is independently closed or open with probability p and 1  p, respec/vely. If a signal is fed to the input, what is the probability that it is transmi6ed to the output? Example 4.16b: Parallel circuits If a circuit is composed only of parallel components, then what is the probability that, at a speciﬁed /me, the system is working? Example: Bayes’s Rule In a bolt factory, 30, 50, and 20% of the produc/on is manufactured by machines I, II, and III, respec/vely. If 4, 5, and 3% of the output of these respec/ve machines is defec/ve, what is the probability that a randomly selected bolt that is found to be defec/ve is manufactured by machine III? Example: Bayes’s Rule (Monty Hall Problem) This follows the game show ‘Let’s Make a Deal’ which was hosted by Monty Hall for many years. In the game show, there are three doors, behind each of which is one prize. Two of the prizes are worthless and the other one is valuable. A contestant selects one of the doors, following which the game show host (who does know where the valuable prize is), opens one of the remaining two doors to reveal a worthless prize. The host then oﬀers the contestant the opportunity to change his select. Should the contestant switch? Example: Bayes’s Rule A diagnos/c test for a certain disease has a 99% sensi/vity and a 95% speciﬁcity. Only 1% of the popula/on has the disease in ques/on. If the diagnos/c test reports that a person chosen at random from the popula/on tests posi/ve, what is the probability that the person does, in fact, have the disease? ...
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 Spring '08
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 Probability

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