Testing for a disease can be made more efficient by combining samples. If the samples from six people are combined and the mixture tests negati) positive sample will always test positive, no matter how many negative samples it is mixed with. Assuming the probability of a single sample testing 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 (Round to three decimal places as need.

To determine the probability that a mixture of six samples tests positive, we can use the concept of **group testing**. The idea is to combine multiple samples and test them together. If the combined test is negative, all individuals in the group are cleared. If it's positive, further testing is required to identify the positive samples. ### Given: - **Probability of a single sample testing positive (p):** 0.005 - **Number of samples combined (n):** 6 ### Probability Calculation: The probability that the mixture tests **positive** is equal to the probability that **at least one** of the six samples is positive. This can be calculated using the complement rule: \[ P(\text{mixture positive}) = 1 - P(\text{all six samples negative}) \] First, calculate the probability that a single sample is negative: \[ P(\text{single negative}) = 1 - p = 1 - 0.005 = 0.995 \] Since the samples are independent, the probability that all six samples are negative is: \[ P(\text{all six negative}) = (0.995)^6 \] Now, compute \( (0.995)^6 \): \[ (0.995)^6 \approx 0.970 \] Therefore, the probability that the mixture tests positive is: \[ P(\text{mixture positive}) = 1 - 0.970 = 0.030 \] ### Conclusion: - **Probability of a positive mixture test:** **0.030** (rounded to three decimal places) This means there is a **3.0%** chance that a mixture of six samples will test positive. Given this low probability, group testing is quite efficient, and further individual testing is rarely necessary unless the mixture test comes back positive. **Final Answer:** The probability of a positive mixture test is **0.030**.