Testing for a disease can be made more efficient by combining samples. If the samples from four people are combined and the mixture tests positive sample will always test positive, no matter how many negative samples it is mixed with. Assuming the probability of a single sample samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely necessary? The probability of a positive test result is \( \square \). (Round to three decimal places as needed.)

To determine the probability that a pooled sample of four individuals tests positive, we need to consider the probability that **at least one** of the four individual samples is positive. Let's denote: - \( p \) as the probability that a single individual sample is positive. - \( 1 - p \) as the probability that a single individual sample is negative. When you pool four samples together: - The only way the pooled sample tests negative is if **all four** individual samples are negative. - The probability that all four samples are negative is \( (1 - p)^4 \). Therefore, the probability that the pooled sample **tests positive** is the complement of all samples being negative: \[ \text{Probability of positive pooled test} = 1 - (1 - p)^4 \] **Rounded to three decimal places:** \[ \boxed{1 - (1 - p)^4} \] **Example Calculation:** If, for instance, the probability \( p \) of a single sample being positive is 0.05 (5%), then: \[ 1 - (1 - 0.05)^4 = 1 - (0.95)^4 \approx 1 - 0.8145 = 0.1855 \approx 0.186 \] So, the probability of the pooled test being positive would be **0.186** when \( p = 0.05 \). **Conclusion:** Without a specific value for \( p \), the general formula for the probability of a positive pooled test is: \[ \boxed{1 - (1 - p)^4} \] You can plug in the specific probability \( p \) to find the numerical probability rounded to three decimal places.