1. (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?
# The probability that an individual who is reported as positive by the new test actually has the disease is 49.48%.

#A1 = Correct Positive
#A2 = Correct Negative
#B = Positive

A1 <- 0.001
B_A1 <- 0.96
B_A2 <- 0.98
A1_B <- (B_A1*A1)/(B_A1*A1+B_A2*A1)
A1_B
## [1] 0.4948454
# Variables
n <- 100000
positive <- 100000
test <- 1000
prob <- A1_B

# There seems to be some ambiguity 
# Assuming that each of the 100,000 individuals have tested positive

# Cost of test
test_cost <- n*test

# Assuming that n=100,000 is the total population

# Total people with positive case
pos_res <- n*prob

# Total cost for positive cases
pos_cost = pos_res*positive

# Total cost
total_cost <- test_cost + pos_cost
total_cost
## [1] 5048453608
  1. (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?
n <- 24
x <- 2
p <- 0.05

# probability that, after 24 months, you received exactly 2 inspections
dbinom(x,n,p)
## [1] 0.2232381
# probability that, after 24 months, you received 2 or more inspections
# 1 minus the sum of the probabilities of having 1 or 0 inspections.
1-(dbinom(1,n,p)+dbinom(0,n,p))
## [1] 0.3391827
# probability that your received fewer than 2 inspections
# Sum of the probabilities of having 1 or 0 inspections.
dbinom(1,n,p)+dbinom(0,n,p)
## [1] 0.6608173
# expected number of inspections you should have received
# The expected number of inspections is 24*0.05
n*p
## [1] 1.2
# standard deviation
sqrt(n*(p*(1-p)))
## [1] 1.067708
  1. (Poisson). 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? What is the probability that more than 10 arrive in one hour? How many would you expect to arrive in 8 hours? What is the standard deviation of the appropriate probability distribution? 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?

x <-3
lambda <- 10

#probability that exactly 3 arrive in one hour
dpois(x,lambda)
## [1] 0.007566655
#probability that more than 10 arrive in one hour
p = 0
for (x in range(0:10))
{p = p + dpois(x,10)}
1-p
## [1] 0.8748446
#How many would you expect to arrive in 8 hours? 
lambda*8
## [1] 80
#standard deviation of the appropriate probability distribution
sqrt(lambda)
## [1] 3.162278
#percent utilization
80/72*100
## [1] 111.1111

There is over utilization of the services, it would be recommended to increase hours or hire more staff.

  1. (Hypergeometric). 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? How many nurses would we have expected your subordinate to send? How many non-nurses would we have expected your subordinate to send?
#probability he/she would have selected five nurses for the trips
dhyper(5,15,15,6,log=FALSE)
## [1] 0.07586207
# How many nurses would we have expected your subordinate to send
n <- 6*15/30

# How many non-nurses would we have expected your subordinate to send
nn <- n-3
  1. (Geometric). 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? In the course of 15 months? What is the expected number of hours that a driver will drive before being seriously injured? Given that a driver has driven 1200 hours, what is the probability that he or she will be injured in the next 100 hours?
#probability that the driver will be seriously injured during the course of the year
q <- 1200
prob <- 0.001
1-pgeom(q,prob)
## [1] 0.3007124
# In the course of 15 months
q <- 1500
1-pgeom(q, prob)
## [1] 0.2227398
#expected number of hours that a driver will drive before being seriously injured
1/0.001
## [1] 1000
#probability that he or she will be injured in the next 100 hours
pgeom(100, 0.001)
## [1] 0.09611265
  1. 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?
    What is the expected value?
# Poisson Distribution

#probability that the generator will fail more than twice in 1000 hours
1-ppois(2,1)
## [1] 0.0803014
#expected value
1
## [1] 1
  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? 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? What is the expected waiting time?
#Uniform Distribution

#probability that this patient will wait more than 10 minutes
1-punif(10,0,30)
## [1] 0.6666667
# If the patient has already waited 10 minutes, what is the probability that he/she will wait at least another 5 minutes
(1-punif(15,0,30))/(1 - punif(10,0,30))
## [1] 0.75
#expected waiting time, average waiting time
sum(0,30)/2
## [1] 15
  1. 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? What is the standard deviation? What is the probability that your MRI will fail after 8 years? 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?
#Exponential Distribution

#expected failure time
10
## [1] 10
#Standard deviation
1/0.1
## [1] 10
#probability that your MRI will fail after 8 years
1-pexp(8,0.1)
## [1] 0.449329
#what is the probability that it will fail in the next two years
pexp(10,.1)-pexp(8,.1)
## [1] 0.08144952