Testing for a disease can be made more efficient by combining samples. If the samples from six people are combined and the mixture tests​ negative, then all six 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 six samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely​ necessary?

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Testing for a disease can be made more efficient by combining samples. If the samples from

six

people are combined and the mixture tests​ negative, then

all six

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 six samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely​ necessary?

Expert Solution
Step 1

Given Information:

Probability of a single sample testing positive is 0.2

Probability of a single sample testing negative is 1-0.2=0.8

We only save time when all six are negative, which has a probability of (0.8)6=0.262144

 

combined sample tests negative = 1 with probability 0.262144

Combined sample tests positive = 1 + 6 retests = 7 with probability 1-0.262144=0.737856

Expected number of tests is obtained as follows:

EX=inipi/n=1*0.262144+7*0.7378566=0.9045226667<1

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