Problem

\begin{tabular}{|l|c|c|} & Positive Result & Negative Result \\ \hline Disease Present & 103 & 15 \\ \hline Disease Not Present & 17 & 207 \\ \hline \end{tabular} 23. Determine the probability that the blood test will detect the disease, if you have the disease.

Solution

Step 1 :Given the table, we can see that there are 103 cases where the blood test detected the disease and the disease was actually present, and 15 cases where the disease was present but the blood test did not detect it. So, the total number of cases where the disease is present is 103 + 15 = 118.

Step 2 :The question is asking for the probability that the blood test will detect the disease given that you have the disease. This is a conditional probability problem. The formula for conditional probability is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\). In this case, event A is the blood test detecting the disease and event B is having the disease.

Step 3 :\(P(A \cap B)\) is the probability of both events happening, which is the number of positive results when the disease is present. In this case, \(P(A \cap B)\) = 103.

Step 4 :\(P(B)\) is the probability of having the disease, which is the total number of cases where the disease is present. In this case, \(P(B)\) = 118.

Step 5 :Substituting these values into the formula, we get \(P(A|B) = \frac{103}{118} = 0.8728813559322034\).

Step 6 :Final Answer: The probability that the blood test will detect the disease, if you have the disease is approximately \(\boxed{0.873}\).

From Solvely APP
Source: https://solvelyapp.com/problems/17494/

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