Question

a pharmaceutical company is developing a new test for gluten intolerance. the company is working to lower the \false positive rate\ which is currently 25%. a false positive test result occurs when the test indicates someone is gluten intolerant, but in reality they are not. if 2 volunteers were randomly selected to take the test, what is the probability that at most 1 will receive a false positive result? write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Calculate the probability of the complementary event

The complementary event of "at most 1 false - positive" is "2 false - positives". Let \(p = 0.25\) (probability of a false - positive) and \(q=1 - p=0.75\) (probability of a non - false - positive).
The probability of 2 false - positives is \(P(X = 2)=p\times p\) (using the multiplication rule for independent events).

$$P(X = 2)=0.25\times0.25=0.0625$$

Step2: Calculate the probability of at most 1 false - positive

We use the formula \(P(X\leq1)=1 - P(X = 2)\)

$$P(X\leq1)=1-0.0625 = 0.9375$$

Answer:

\(0.938\)