Question

Testing for a disease can be made more efficient by combining samples. If the samples from two people are combined and the mixture tests​ negative, then both 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 0.1​, find the probability of a positive result for two 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 nothing.

Answer 1

P(positive test result) = 1 - P(negative test result)

= 1 - P(all 5 samples test negative)

=1-(1-0.1)⁵

=1-(0.9)⁵

=1-0.59049

=0.40951

=0.410

note: if 0.410 do not  accept please try with 0.409. (rounding up to three decimal)

Answer 2

The probability that a single sample testing positive is 0.21 Therefore, the probability that a single sample testing negative is 1 - 0.1 = 0.9

The sample would be positive if at least one positive sample is mixed.

Therefore, the probability of a positive test result is:

P(positive) = 1 - P(all negative)

= 1 - 0.9^2

= 0.19