Question

Assuming the probability of a single sample testing positive is 0.15​, find the probability of a positive result for two samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely​ necessary?

Answer 1

Answer:

$P(Positive\ Mixture) = 0.2775$

The probability is not low

Step-by-step explanation:

Given

$P(Single\ Positive) = 0.15$

$n = 2$

Required

$P(Positive\ Mixture)$

First, we calculate the probability of single negative using the complement rule

$P(Single\ Negative) = 1 - P(Single\ Positive)$

$P(Single\ Negative) = 1 - 0.15$

$P(Single\ Negative) = 0.85$

$P(Positive\ Mixture)$ is calculated using:

$P(Positive\ Mixture) = 1 - P(All\ Negative)$ ---- i.e. complement rule

So, we have:

$P(Positive\ Mixture) = 1 - 0.85^2$

$P(Positive\ Mixture) = 1 - 0.7225$

$P(Positive\ Mixture) = 0.2775$

Probabilities less than 0.05 are considered low.

So, we can consider that the probability is not low because 0.2775 > 0.05