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?
Those that test positive are the 96% and the 2% that get false positive. Also note that .001% of 100000 is actually positive, which is 100.
That means out of the 100 that actually has the disease 96 people, plus 2% of the 999,900 that tests positive, but don’t actually have it. So, 96 + 1998 = 2094 people that test positive. Which means 2094/96 = 4.5% are actually positive.
Using the Bayesian formula it would be the probability of testing positive percentage P(Positive)=.96 by the percentage of population that actually has it P(Pos_Rate) = .001 divided by the P(Positive)P(Pos_Rate) plus P(False_Pos)*P(Pop_Neg) = .0458.
\[ P(Actually\_Positive) = \frac{P(Positive)P(Pos\_Rate)}{P(Positive)P(Pos\_Rate)+P(False\_Pos)*P(Neg\_Rate)}\]
#Probabilities of: positive, false_negative, negative and false positive
positive<-.96
false_negative <- 1-positive
negative <- .98
false_positive <- 1-negative
#Positive and Negative Rates
pos_rate<- .001
neg_rate <- 1 - pos_rate
#Population total, positve, negative
population=100000
pop_pos <- population*pos_rate
pop_neg <- (population-pos_rate)*neg_rate
#Costs
med_cost<-100000
test_cost<-1000
P_actually_has <- (positive*pos_rate)/((positive*pos_rate)+(false_positive*neg_rate))
print(paste0("Percentage that actually is positive: ",round(P_actually_has*100,2),"%"),collapse='')## [1] "Percentage that actually is positive: 4.58%"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?
## [1] "Initial testing costs: $1e+08"## [1] "Number of retests: 2098"## [1] "Cost of retesting: $2098000"## [1] "Total testing costs: $102098000"## [1] "One year Medical Costs for positive: 1e+07"## [1] "The total for testing and one year of care is: $112098000"\[P\left(x;p,n\right) = \binom{n}{k} \left(p\right)^x\left(1-p\right)^{\left(n-k\right)}\hspace{.5cm}for\hspace{.15cm}x = 0,1,2,...,n\] \[ where:\hspace{.2cm} \binom{n}{k} = \frac{n!}{k!(n-k)!}\]
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?
## [1] "There is a 22.3% chance of getting 2 inspectitions in 24 months."What is the probability that, after 24 months, you received 2 or more inspections?
## [1] "The probability of getting more than 2 inspections in 24 months is: 33.9%"What is the probability that your received fewer than 2 inspections?
## [1] "The probability of getting less than two inspections is: 66.1%"What is the expected number of inspections you should have received?
## [1] "The exepected number of visits is 1.2"What is the standard deviation?
Standard deviation of a binomial is \(\sqrt{n*p*(1-p)}\)
## [1] "The standard deviation is: 1.06770782520313"You are modeling the family practice clinic and notice that patients arrive at a rate of 10 per hour.
What is the probability that exactly 3 arrive in one hour?
## [1] "The probability that exactly 3 arrive in one hour is 0.0076"What is the probability that more than 10 arrive in one hour?
## [1] "The probability that more than 10 arrive in one hour is 0.417"How many would you expect to arrive in 8 hours?
I would expect 80 since a rate of 10 per hour come in.
What is the standard deviation of the appropriate probability distribution?
## [1] "The standard deviation is 3.162"If there are three family practice providers that can see 24 templated patients each day, what is the percent utilization and what are your recommendations?
Individually they do 30% of the business that the one clinic does. Seeing how it says it can see 24 patients each day then I am assuming that the number of doctors at each clinic and the size of each office is quite lower. If that is the case then I would recommend increasing space and adding a doctor or two. Or possibly combining the three clinics into one where they can increase space and cut costs.
Your subordinate with 30 supervisors was recently accused of favoring nurses. 15 of the subordinate’s workers are nurses and 15 are other than nurses. As evidence of malfeasance, the accuser stated that there were 6 company-paid trips to Disney World for which everyone was eligible. The supervisor sent 5 nurses and 1 non-nurse.
If your subordinate acted innocently, what was the probability he/she would have selected five nurses for the trips?
## [1] "The probability of innocently picking 5 nurses for the trips would be 0.076"How many nurses would we have expected your subordinate to send?
I would have expected 3
How many non-nurses would we have expected your subordinate to send?
I would have expected 3
The probability of being seriously injured in a car crash in an unspecified location is about .1% per hour. A driver is required to traverse this area for 1200 hours in the course of a year.
What is the probability that the driver will be seriously injured during the course of the year?
## [1] "The probability of a driver getting seriously injured while driving during the course of the year is 30.07%"In the course of 15 months?
## [1] "The probability of a driver getting seriously injured while driving during the course of 15 months is 22.27%"What is the expected number of hours that a driver will drive before being seriously injured?
## [1] "The expected number of hours that will be driven before injury is 1000"Given that a driver has driven 1200 hours, what is the probability that he or she will be injured in the next 100 hours?
## [1] "The probability that a driver will be injured in the next 100 hours is 9.611"You are working in a hospital that is running off of a primary generator which fails about once in 1000 hours.
What is the probability that the generator will fail more than twice in 1000 hours?
## [1] "The probability that the generator will fail more than twise is 0.08"What is the expected value?
The expected value is \(\lambda\) which is 1
A surgical patient arrives for surgery precisely at a given time. Based on previous analysis (or a lack of knowledge assumption), you know that the waiting time is uniformly distributed from 0 to 30 minutes.
What is the probability that this patient will wait more than 10 minutes?
## [1] "The probability of waiting more than 10 minutes is 66.7%"If the patient has already waited 10 minutes, what is the probability that he/she will wait at least another 5 minutes prior to being seen?
## [1] "The probability of waiting at least 5 more minutes is 25%"What is the expected waiting time?
The expected wait time is 15 minutes.
Your hospital owns an old MRI, which has a manufacturer’s lifetime of about 10 years (expected value). Based on previous studies, we know that the failure of most MRIs obeys an exponential distribution.
What is the expected failure time?
10 years, same as the warranty.
What is the standard deviation?
1/10 is the standard deviation
## [1] 0.1What is the probability that your MRI will fail after 8 years?
## [1] "The probability that it will fail after 8 years is 0.45"Now assume that you have owned the machine for 8 years. Given that you already owned the machine 8 years, what is the probability that it will fail in the next two years?
## [1] "The probability that it will fail in the next two years is 0.081"