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Schizophrenia affects about 1% of the population. Imagine that scientists discover a simple blood test that can tell with 95% accuracy whether a person will develop schizophrenia. That is, 95% of the people who actually will develop schizophrenia will correctly test positive (but 5% will falsely test negative), and 95% of the people who actually will never develop schizophrenia will correctly test negative (but 5% will falsely test positive). You take the test and the results come back positive. What is the approximate probability that you will actually develop schizophrenia?

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Answer:

If you test positive, the approximate probability that you will actually develop schizophrenia is 16.10%.

Step-by-step explanation:

We have these following probabilities:

A 1% probability of having Schizophrenia.

A 99% of not having Schizophrenia.

If a person has Schizophrenia, a 95% probability of testing positive.

If a person does not have Schizophrenia, a 5% probability of testing positive.

What is the approximate probability that you will actually develop schizophrenia?

This can be formulated as the following problem:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

So:

What is the probability of you developing schizophrenia, given that you tested positive?

P(B) is the probability of the person having the disease. So [tex]P(B) = 0.01[/tex]

P(A/B) is the probability of the person being diagnosticated, given that she has the disease. So [tex]P(A/B) = 0.95[/tex].

P(A) is the probability of the person being diagnosticated. That is 95% of 1% and 5% of 99%. So

[tex]P(A) = 0.01(0.95) + 0.99(0.05) = 0.059[/tex]

Finally

[tex]P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.01*0.95}{0.059} = 0.1610[/tex]

If you test positive, the approximate probability that you will actually develop schizophrenia is 16.10%.

fichoh

Using the Naive Bayes rule ; the probability that a tested person actually develops schizophrenia given that test result comes back positive is 0.161

Naive Bayes rule :

  • [tex]P(A|B) = \frac{P(B|A) * P(A)}{P(B)}[/tex]

  • P(A|B) = P of A being True given that B is True

  • P(A) = probability of A

  • P(B|A) = P of B bring true given that A is True

  • P(B) = probability of B

Let :

  • P(A) = Probability of Schizophrenia = 1% = 0.01

  • P(B) = probability of True positive result

  • P(A|B) =?

  • P(B|A) = P of positive result given person has schizophrenia = 95% = 0.95

The probability of obtaining a positive result :

P(B) = [P(B|A) × P(A)] + [P(B|A)' × P(A)']

P(B) = (0.95 × 0.01) + (0.05 × 0.99)

P(B) = 0.059

Hence,

[tex]P(A|B) = \frac{0.95 \times 0.01}{0.059} = \frac{0.0095}{0.059} = 0.161[/tex]

Therefore, the probability of schizophrenia given a true positive result is 0.161

Learn more :https://brainly.com/question/18405415

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