Use Bayes' rule to find the indicated probability. The incidence of a certain disease in the town of Springwell is \( 4 \% \). A new test has been developed to diagnose the disease. Using this test, \( 91 \% \) of those who have the disease test positive while \( 4 \% \) of those who do not have the disease test positive ("false positive"). If a person tests positive, what is the probability that he or she actually has the disease? 0. 0.487 00.91 00.865 00.527 0.438

To find the probability that a person actually has the disease given that they test positive, we can use Bayes' rule. Let's denote: - \( P(D) \) as the probability of having the disease (4% or 0.04), - \( P(\text{Test Positive}|D) \) as the probability of testing positive given that the person has the disease (91% or 0.91), - \( P(\text{Test Positive}|\text{No Disease}) \) as the probability of testing positive given that the person does not have the disease (4% or 0.04), - \( P(\text{No Disease}) \) as the probability of not having the disease (96% or 0.96). We want to find \( P(D|\text{Test Positive}) \), the probability of having the disease given that the person tests positive. Using Bayes' rule, we have: \[ P(D|\text{Test Positive}) = \frac{P(\text{Test Positive}|D) \cdot P(D)}{P(\text{Test Positive}|D) \cdot P(D) + P(\text{Test Positive}|\text{No Disease}) \cdot P(\text{No Disease})} \] Substitute the given values: \[ P(D|\text{Test Positive}) = \frac{0.91 \cdot 0.04}{0.91 \cdot 0.04 + 0.04 \cdot 0.96} \] Now, we can calculate the probability. Calculate the value by following steps: - step0: Calculate: \(\frac{0.91\times 0.04}{\left(0.91\times 0.04+0.04\times 0.96\right)}\) - step1: Remove the parentheses: \(\frac{0.91\times 0.04}{0.91\times 0.04+0.04\times 0.96}\) - step2: Multiply the numbers: \(\frac{0.91\times 0.04}{0.0364+0.04\times 0.96}\) - step3: Multiply the numbers: \(\frac{0.91\times 0.04}{0.0364+0.0384}\) - step4: Multiply the numbers: \(\frac{0.0364}{0.0364+0.0384}\) - step5: Add the numbers: \(\frac{0.0364}{0.0748}\) - step6: Convert the expressions: \(\frac{\frac{91}{2500}}{\frac{187}{2500}}\) - step7: Multiply by the reciprocal: \(\frac{91}{2500}\times \frac{2500}{187}\) - step8: Reduce the numbers: \(91\times \frac{1}{187}\) - step9: Multiply: \(\frac{91}{187}\) The probability that a person actually has the disease given that they test positive is approximately 0.487 or 48.7631%.