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 if further testing is rarely necessary, let’s first clarify the probability of a positive test result when samples are combined. If the probability of a single sample testing positive is \( p \), you can find the probability of the combined sample testing negative, which is \( (1 - p)^4 \) for four samples. Thus, the probability of the combined sample testing positive would be: \[ P(\text{positive}) = 1 - (1 - p)^4 \] If \( p \) is low enough (for example, let’s say \( p = 0.1\)), then: \[ P(\text{positive}) = 1 - (0.9)^4 \approx 1 - 0.6561 = 0.3439 \] Rounded to three decimal places, this gives \( P(\text{positive}) = 0.344 \). If further testing of individuals is conducted rarely, you'd generally want this probability to be less than 0.1, suggesting you're combining samples effectively. With lower \( p \), the benefit is even more pronounced!