Binomial Method

Since we have the variables N=30 pi=0.05 The probability statement can be written as P(X>=5|N=30,pi=0.05) or 1-P(X<=4|N=30,pi=0.05)

N<-30
pi<-0.05
sum(dbinom(5:N,N,pi))
## [1] 0.01563551
1-pbinom(4,N,pi)
## [1] 0.01563551

Binomial model: The probability that this team of surgeons would experience 5 or more deaths in 30 trails if they were operating at the national rate of 0.05 is 0.01563551

Poisson Method

Probability of procedure resulting in death = 0.05 or 5 out of each 100 procedures. lambda=5 P(Y>=5|lambda=5,t=30/100) or as 1-P(Y<=4|lambda=5,t=30/100)

lambda<-5
t<-30/100
sum(dpois(5:100000,lambda*t))
## [1] 0.01857594
1-ppois(4,lambda*t)
## [1] 0.01857594

Poisson model: The probability that this team of surgeons would experience 5 or more deaths in 30 trails if they were operating at the national rate of 0.05 is 0.01857549