Question

testing for a disease can be made more efficient by combining samples. if the samples from three people are combined and the mixture tests negative, then all three samples are negative. on the other hand, one positive sample will always test positive, no matter how many negative samples it is mixed with. assuming the probability of a single sample testing positive is 0.2, find the probability of a positive result for three 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 0.488. (round to three decimal places as needed.)
is the probability low enough so that further testing of the individual samples is rarely necessary?
a. the probability is low, so further testing of the individual samples will be a rarely necessary event.
b. the probability is not low, so further testing will not be necessary for any of the mixtures.
c. the probability is not low, so further testing of the individual samples will frequently be a necessary event.
d. the probability is low, so further testing will be necessary for all of the combined mixtures.

Explanation:

Step1: Find probability of negative result for all three samples

The probability of a single - sample testing negative is \(1 - 0.2=0.8\). Since the samples are independent, the probability that all three samples are negative is \(P(\text{all negative})=(0.8)\times(0.8)\times(0.8)=0.512\).

Step2: Find probability of positive result for the combined sample

The probability of a positive result for the combined sample is the complement of the probability that all three samples are negative. So \(P(\text{positive result}) = 1 - P(\text{all negative})=1 - 0.512 = 0.488\).

Step3: Determine if the probability is low

A probability of \(0.488\) is not considered low. A low probability is usually something much smaller, like less than \(0.1\) or \(0.05\). Since the probability of a positive result for the combined sample is \(0.488\), further testing of the individual samples will frequently be a necessary event.

Answer:

C. The probability is not low, so further testing of the individual samples will frequently be a necessary event.