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?

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.

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

And is the probability of X happening.

A number of sucesses x is considered unusually low if and unusually high if

In this problem, we have that:

There are four samples, so

Each sample has a probability of 0.1 of being positive, so .

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 . Either the number of positive samples is 0, or it is greater than 0. The sum of the probabilities is decimal 1. So:

In which

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.

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step-by-step explanation: