Respuesta :
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.
[tex]P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And [tex]\pi[/tex] is the probability of X happening.
A number of sucesses x is considered unusually low if [tex]P(X \leq x) \leq 0.05[/tex] and unusually high if [tex]P(X \geq x) \geq 0.05[/tex]
In this problem, we have that:
There are four samples, so [tex]n = 4[/tex]
Each sample has a probability of 0.1 of being positive, so [tex]\pi = 0.1[/tex].
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 [tex]P(X>0)[/tex]. Either the number of positive samples is 0, or it is greater than 0. The sum of the probabilities is decimal 1. So:
[tex]P(X = 0) + P(X>0) = 1[/tex]
[tex]P(X>0) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}[/tex]
[tex]P(X = 0) = C_{4,0}.(0.10)^{0}.(0.9)^{4} = 0.6561[/tex]
[tex]P(X>0) = 1 - P(X = 0) = 1 - 0.6561 = 0.3439[/tex]
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.
