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 key probabilities. The problem involves conditional probabilities and complements. The condition affects 1 out of 100 people, so the probability of having the condition, P(B), is 0.01. The complement, P(B'), is the probability of not having the condition, which is 1 - P(B).
Step 2: Calculate P(B'). Since P(B) = 0.01, the complement is P(B') = 1 - 0.01. This represents the probability that a person does not have the condition.
Step 3: Identify P(A|B). This is the probability that a person tests positive (event A) given that they have the condition (event B). The problem states that the test is correct 95% of the time for those with the condition, so P(A|B) = 0.95.
Step 4: Identify P(A|B'). This is the probability that a person tests positive (event A) given that they do not have the condition (event B'). The problem states that the test gives a wrong result 10% of the time for those without the condition, so P(A|B') = 0.10.
Step 5: Summarize the results. The probabilities are P(B') = 0.99, P(A|B) = 0.95, and P(A|B') = 0.10. These values can now be used for further analysis or decision-making.
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