A test for the presence of a disease has probability 20 of...

A test for the presence of a disease has probability 20 of giving a false-positive reading (indicating that an individual has the disease when this is not the case) and probability. 10 of giving a false-negative result. Suppose that ten individuals are tested, five of whom have the disease and five of whom do not. Let $X=$ the number of positive readings that result. (a) Does $X$ have a binomial distribution? Explain your reasoning. (b) What is the probability that exactly three of the ten test results are positive?

A test for the presence of a disease has probability 20 of giving a false-positive reading (indicating that an individual has the disease when this is not the case) and probability. 10 of giving a false-negative result. Suppose that ten individuals are tested, five of whom have the disease and five of whom do not. Let $X=$ the number of positive readings that result.
(a) Does $X$ have a binomial distribution? Explain your reasoning.
(b) What is the probability that exactly three of the ten test results are positive?

Key Concepts

Poisson Binomial Distribution

When the trials are independent but the probability of success varies from one trial to another, as is the case when different subgroups have different testing accuracies, the total number of successes follows a Poisson binomial distribution rather than a true binomial distribution. This distribution accounts for the non-identical nature of the probabilities, making it more appropriate for situations involving heterogeneous groups.

Binomial Distribution

A binomial distribution models the number of successes in a series of independent trials where each trial has exactly two outcomes (success or failure) and the probability of success is constant across all trials. It relies on the assumptions of a fixed number of trials, independent outcomes, and identically distributed success probabilities.

False Positives and False Negatives

False positives and false negatives are errors inherent in diagnostic testing. A false positive occurs when a test wrongly indicates the presence of a condition in someone who does not have it, while a false negative occurs when the test fails to detect a condition that is present. These metrics are critical in determining the effective probability of a positive test result, especially when different groups have different underlying conditions affecting these error rates.

Transcript

00:02 We're told that a test for the presence of a disease has a probability of 20 % of giving a false positive reading.

00:12 So this means that an individual has the disease and this is not the case.

00:17 And a probability 0 .10 of giving a false negative reading.

00:23 One of their words that an individual does not have the disease when in fact this is not the case.

00:35 They do have the disease.

00:39 Now, we're told that 10 individuals are tested, five of whom have the disease and five with whom do not.

00:47 And we're given that x is the random variable, which represents the number of positive readings that result.

00:54 In part a, we're asked to determine if x has a binomial distribution and to explain.

01:05 While we think about criteria for a binomial distribution, so we require that the binomial experiment, the trials would have to be independent, and the probability of success would have to be the same for all the tests.

01:24 However, we have that this is not the case, because the probability of success is not the same for all the tests.

02:05 And in part b, we're asked to find the probability that exactly three of the 10 test results are positive.

02:13 So this part really shows that even though this may not have a binomial distribution, it's still possible to calculate probabilities using their given information.

02:27 So we have there are four ways that exactly three people could have positive results.

03:06 So we'll list these different, four different combinations.

03:11 Well, we have that on the one hand, the number of people who have the disease is zero.

03:24 Number of people who don't have the disease d prime, this is three, and yet all three of these people have positive results.

03:42 We have the probability of this outcome.

03:46 This is five, choose zero people actually have the disease, times the probability that they have the disease, 0 to the 0, times the probability of a false positive, which is 0 .8 to the 5 minus 0 or 5th power times, again we're choosing from out of 5 people so that of the 5 people who don't have the disease, we choose three of them, and the probability choosing somebody who does not have the disease, or i guess i'm giving a false negative, this is 0 .1.

05:14 So assuming we get three false negatives.

05:30 Sorry.

05:37 Probability of there being an actual positive result is 0 .9.

05:58 So you get 0 .9 cubed times 0 .1 squared.

06:03 Point 1 is the probability of giving a false negative.

06:10 So the 0 .9 is probability if they're not being a false negative.

06:16 So that's one case.

06:18 And in fact, we can calculate this first expression to be .32768, and the second expression is calculated to be .0729 approximately, and together this is approximately .02389.

06:50 Now another combination that's possible is we could have one person who actually has the disease and two people who don't have the disease.

07:06 So, if we can calculate the probability of this outcome, we have, there are five people who actually have the disease, of which we're picking one, times, and the probability of a false positive, which is 0 .2, the first power, times the probability of not having a false positive, which is 0 .8, and to the 5 minus 1 is 4.

08:02 So there are meaning five, don't want to false positive times.

08:10 And then for the ones, don't know the disease.

08:13 Well, there are five people who belong in the disease...

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