Testing for a disease can be made more efficient by...

Testing for a disease can be made more efficient by combining samples. If the samples from four people are combined and the mixture tests negative, then all four 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 0.15, find the probability of a positive result for four 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 needed.)

Transcript

00:01 Okay, in this question, you're given the testing for diseases, and they're talking about trying to come up with a more efficient practice of combining samples so that they don't have to test as many samples.

00:13 So what they're saying is that they combine four people's samples into a mixture and then test that mixture.

00:22 If the result from that mixture shows up negative, then all four people were negative.

00:30 Even if one person in that group of four was positive, that mixture of four samples will show up positive.

00:42 Okay, so that's the situation or scenario we're dealing with here.

00:49 So the question's asking us, and they give us the probability that any single test would be positive of 0 .15.

00:58 We're asked to find the probability that you would get a positive result when you mix four samples, given the probability of a single sample being 0 .15.

01:13 So if you think about this question, there's a bunch of different combinations of positive results in the group of four that could lead to an overall positive result in the mixture.

01:28 I'm just going to lay out some of those here.

01:31 So again, we have four people here, and i'm talking about the likelihood that the mixture is going to turn up positive.

01:42 It can turn up positive if one person is positive.

01:46 So if i had a positive, negative, negative, and negative person in my sample, the whole mixture is turning up positive.

01:54 If i had a negative person and then a positive person, then a negative and a negative, the whole sample is turning up positive.

02:02 And i could keep going with that, moving my positive person into the different slots here.

02:08 We could also have two positive patients, two negatives.

02:13 That would turn up.

02:14 So as you can see, what i'm getting at here is that there would be a lot of different ways that that four sample mixture would turn up positive.

02:24 And that's what we're trying to figure out the probability of.

02:26 We would have to find all those different individual scenarios for making the mixture come up positive.

02:37 But there's an easier approach.

02:39 If you think of this as a complement situation where the complement of having at least one positive in the four sample, the complement of that would be having no positives...

Question

Please give Ace some feedback