4. Probability
Bayes' Theorem
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A rare condition affects 1 out of every 100 people. The test for this condition has the following probabilities: If a person has the condition, the test is correct 95% of the time. If a person does not have the condition, the test gives a wrong result 10% of the time. If A is the event 'tested positive' and B is the event 'has condition,' find P(B'), P(AIB), and P(A|B').
A
P(B') = 0.99; P(A|B) = 0.95; P(A|B') = 0.10
B
P(B') = 0.99; P(A|B) = 0.95; P(A|B') = 0.01
C
P(B') = 0.95; P(A|B) = 0.99; P(A|B') = 0.01
D
P(B') = 0.95; P(A|B) = 0.99; P(A|B') = 0.10
Verified step by step guidance1
Step 1: Understand the problem and identify the given probabilities. The condition affects 1 out of 100 people, so P(B) = 0.01 (probability of having the condition) and P(B') = 1 - P(B) = 0.99 (probability of not having the condition).
Step 2: Recognize the test's accuracy. If a person has the condition, the test is correct 95% of the time, so P(A|B) = 0.95 (probability of testing positive given the person has the condition).
Step 3: Understand the test's error rate for people without the condition. If a person does not have the condition, the test gives a wrong result 10% of the time, so P(A|B') = 0.10 (probability of testing positive given the person does not have the condition).
Step 4: Use the complement rule to confirm P(B') = 0.99, as it represents the probability of not having the condition.
Step 5: Summarize the probabilities: P(B') = 0.99, P(A|B) = 0.95, and P(A|B') = 0.10. These values are derived directly from the problem's description and the complement rule.
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