Bayes Theorem What does Bayes Formula helps to
Baye’s Theorem
What does Bayes’ Formula helps to find? – Helps us to find: by having already known
Example-An electronics store sells DVD players made by one of two brands. Customers can also purchase extended warranties (保证) for the DVD player. The following probabilities are given: Let B 1 = event that brand 1 is purchased B 2 = event that brand 2 is purchased E = event that extended warranty is purchased P(B 1) =. 7 P(B 2) =. 3 P(E|B 1) =. 2 P(E|B 2) =. 4
If a DVD customer is selected at random, what is the probability that they purchased the extended warranty? P(B 1) =. 7 P(B 2) =. 3 P(E|B 1) =. 2 P(E|B 2) =. 4 This can happen in one of two ways: 1) They purchased the extended warranty and Brand 1 DVD player OR 2) They purchased the extended warranty and Brand 2 DVD player
P(B 1) =. 7 P(B 2) =. 3 P(E|B 1) =. 2 P(E|B 2) =. 4 These are disjoint events
P(B 1) =. 7 P(B 2) =. 3 P(E|B 1) =. 2 P(E|B 2) =. 4 P(E) = (. 2)(. 7) + (. 4)(. 3) =. 26 This is an example of the Law of Total Probabilities.
Law of Total Probabilities If B 1 and B 2 are disjoint events with probabilities P(B 1) + P(B 2) = 1, for any event E
Law of Total Probabilities More generally B 1, B 2, …, Bk are disjoint events with probabilities P(B 1) + P(B 2) + … + P(Bk) = 1, for any event E
Bayes Rule (Theorem) • A formula discovered by the Reverend Thomas Bayes, to solve what he called “converse” problems. Let’s examine the following problem before looking at the formula. . .
Lyme’s disease (莱姆病) is the leading tick-borne disease in the United States and England. Diagnosis of the disease is difficult and is aided by a test that detects particular antibodies in the blood. The article, “Laboratory Consideration in the Diagnosis and Management of Lyme Borreliosis”, American Journal of Clinical Pathology, 1993, used the following notations: + represents a positive result on a blood test - represents a negative result on a blood test L represents the patient actually has Lymes LC represents the patient doesn’t have Lymes
The article gave the following probabilities: P(L) =. 00207 P(LC) =. 99723 P(+|L) =. 937 P(-|L) =. 063 P(+|Lc) =. 03 P(-|LC) =. 97 Bayes’s converse problem poses this question: “Given that a patient test positive, what is the probability that he or she really has the disease? ” written: P(L|+) This question is of primary concern in medical diagnosis problems!
Lyme’s Disease Continued. . . The article gave the following probabilities: Using the Law of P(L Total Probabilities, C) =. 99723 P(L) =. 00207 P(+|L) =. 937 denominator P(-|L) =becomes. 063 C)P(LC). P(+|L)P(L) + P(+|L C C P(+|L ) =. 03 P(-|L ) =. 97 the Bayes reasoned as follows: Substitute values: Since we can use P(+|L) × P(L) for the numerator.
P(L) =. 00207 P(+|L) =. 937 P(+|LC) =. 03 P(LC) =. 99723 P(-|L) =. 063 P(-|LC) =. 97
Bayes Rule (Theorem) If B 1 and B 2 are disjoint events with probabilities P(B 1) + P(B 2) = 1, for any event E
Bayes Rule (Theorem) More generally B 1, B 2, …, Bk are disjoint events with probabilities P(B 1) + P(B 2) + … + P(Bk) = 1, for any event E
Home work: Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie's wedding?
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