CALCU-The probability of having a certain medical

The probability of having a certain medical condition C1 is p throughout the population. For people with C1, if they do test T1 then the probability of a positive test result is q. For people without C1, if they do test T1 then the probability of a positive test result is r. If someone tested positive in the first test T1 for C1 but now tests negative for C1 from test T1 then what is the probability that they have C1? p=0.03, q=0.8 r=0.2

Solution by Steps

step 1

We need to find the probability that someone has condition

C1

given that they tested positive in the first test

T1

and then tested negative in the second test

T1

. We will use Bayes' Theorem for this calculation

step 2

Let

P(C1) = p = 0.03

,

P(T1^+ | C1) = q = 0.8

, and

P(T1^+ | \neg C1) = r = 0.2

step 3

First, we calculate the probability of testing positive in the first test

T1

:

P(T1^+) = P(T1^+ | C1)P(C1) + P(T1^+ | \neg C1)P(\neg C1)

P(T1^+) = (0.8)(0.03) + (0.2)(0.97)

P(T1^+) = 0.024 + 0.194 = 0.218

step 4

Next, we calculate the probability of testing negative in the second test

T1

given that the first test was positive:

P(T1^- | T1^+) = \frac{P(T1^+ \cap T1^-)}{P(T1^+)}

Since

P(T1^+ \cap T1^-) = P(T1^+ \cap \neg T1^+) = 0

, we need to consider the conditional probabilities:

P(T1^- | T1^+ \cap C1) = 1 - P(T1^+ | C1) = 1 - 0.8 = 0.2

P(T1^- | T1^+ \cap \neg C1) = 1 - P(T1^+ | \neg C1) = 1 - 0.2 = 0.8

step 5

Now, we use Bayes' Theorem to find

P(C1 | T1^+ \cap T1^-)

:

P(C1 | T1^+ \cap T1^-) = \frac{P(T1^+ \cap T1^- | C1)P(C1)}{P(T1^+ \cap T1^-)}

P(T1^+ \cap T1^- | C1) = P(T1^+ | C1)P(T1^- | T1^+ \cap C1) = 0.8 \times 0.2 = 0.16

P(T1^+ \cap T1^-) = P(T1^+ \cap T1^- | C1)P(C1) + P(T1^+ \cap T1^- | \neg C1)P(\neg C1)

P(T1^+ \cap T1^- | \neg C1) = P(T1^+ | \neg C1)P(T1^- | T1^+ \cap \neg C1) = 0.2 \times 0.8 = 0.16

P(T1^+ \cap T1^-) = (0.16)(0.03) + (0.16)(0.97) = 0.0048 + 0.1552 = 0.16

P(C1 | T1^+ \cap T1^-) = \frac{0.16 \times 0.03}{0.16} = 0.03

Answer

The probability that they have

C1

given that they tested positive in the first test

T1

and then tested negative in the second test

T1

is

0.03

.

Key Concept

Bayes' Theorem

Explanation

Bayes' Theorem allows us to update the probability of a hypothesis based on new evidence. In this case, it helps us find the probability of having condition

C1

given the test results.

there is some mistake

I apologize for any confusion. Let's revisit the calculation using Bayes' Theorem to find the probability that someone has condition

C1

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