Question

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 0.1​, 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?

Answer 1

Answer:

There is a 34.39% probability of a positive result for four samples combined into one mixture.

This probability is not low enough so that further testing of the individual samples is rarely​ necessary,

Step-by-step explanation:

There are only two possible outcomes. Either a sample tests positive, or it does not, so we use the binomial probability distribution.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}$

In which $C_{n,x}$ is the number of different combinatios of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And $\pi$ is the probability of X happening.

A number of sucesses x is considered unusually low if $P(X \leq x) \leq 0.05$ and unusually high if $P(X \geq x) \geq 0.05$

In this problem, we have that:

There are four samples, so $n = 4$

Each sample has a probability of 0.1 of being positive, so $\pi = 0.1$.

Assuming the probability of a single sample testing positive is 0.1​, find the probability of a positive result for four samples combined into one mixture.

If any sample is positive, the result of the four samples is positive. So this is $P(X>0)$. Either the number of positive samples is 0, or it is greater than 0. The sum of the probabilities is decimal 1. So:

$P(X = 0) + P(X>0) = 1$

$P(X>0) = 1 - P(X = 0)$

In which

$P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}$

$P(X = 0) = C_{4,0}.(0.10)^{0}.(0.9)^{4} = 0.6561$

$P(X>0) = 1 - P(X = 0) = 1 - 0.6561 = 0.3439$

There is a 34.39% probability of a positive result for four samples combined into one mixture.

This probability if larger than 5%, so it is not low enough so that further testing of the individual samples is rarely​ necessary.