(Bayesian). A new test for multinucleoside-resistant (MNR) human immunodeficiency virus type 1 (HIV-1) variants was recently developed.
The test maintains 96% sensitivity, meaning that, for those with the disease, it will correctly report “positive” for 96% of them.
The test is also 98% specific, meaning that, for those without the disease, 98% will be correctly reported as “negative.”
MNR HIV-1 is considered to be rare (albeit emerging), with about a .1% or .001 prevalence rate. Given the prevalence rate, sensitivity, and specificity estimates, what is the probability that an individual who is reported as positive by the new test actually has the disease?
If the median cost (consider this the best point estimate) is about $100,000 per positive case total and the test itself costs $1000 per administration, what is the total first-year cost for treating 100,000 individuals? \[ \begin{equation} \label{eq:bayes} P(A|B) = \frac{P(B|A)P(A)}{P(B)} \end{equation} \]
Using Bayes’ theorem above, we can plug in values for P(A) and P(B) to calculate the probability of A given B.
Let P(A) be the probability of having the disease. Let P(B) be the probability of testing positive. We want to find the probability of a person having the disease given that they test positive.
\[ \begin{equation} \label{eq:bayes2} P(Disease|Positive) = \frac{P(Positive|Disease)P(Disease)}{P(Positive)} \end{equation} \]
We know P(Disease) is 0.001.
To calculate P(Positive) we must take into consideration the two ways one may get a positive result:
They can either have the disease and test positive, \(P(Positive∣Disease)\) = 0.96
Not have the disease yet get a positive result \(P(Positive∣No Disease)\) = 0.02.
\[ P(Positive)=P(Positive∣Disease)×P(Disease)+P(Positive∣No Disease)×P(No Disease) \] \(P(Disease) = 0.001\)
\(P(NoDisease) = 1-0.001 = 0.999\)
Plugging those numbers in we get: \(0.96 * 0.001 + 0.02 * 0.999\) \(=0.02094\)
Thus: \(\begin{equation} \label{eq:bayes3} P(Disease|Positive) = \frac{0.96 * 0.001}{0.02094} \end{equation}= 0.0458452722 = 4.58\)%
(Binomial). The probability of your organization receiving a Joint Commission inspection in any given month is .05.
What is the probability that, after 24 months, you received exactly 2 inspections?
What is the probability that, after 24 months, you received 2 or more inspections?
What is the probability that your received fewer than 2 inspections?
What is the expected number of inspections you should have received?
What is the standard deviation?
\[P(k) = {n\choose k}p^k(1-p)^{n-k} \]