Binomial Distribution

Model

# With a success constituing as someone who survived the operation, the probability of having 25 out of 30 people survive (5 deaths), with a overall success rate of 95%, is 0.0123
# So based on the National Death Proportion due to this procedure, there is only a 1.23% chance that 25 out of 30 people survive. 

dbinom(25, size = 30, prob = .95)
## [1] 0.01235302
# With there being a 33% chance that 29 out of 30 people survive
dbinom(29, size = 30, prob = .95)
## [1] 0.3389033

Graph

# Shows the probability of each possible outcome, with a success now being that a person died
x <- c(0:30)

y <- c(dbinom(0, size = 30, prob = .05),dbinom(1, size = 30, prob = .05),dbinom(2, size = 30, prob = .05),dbinom(3, size = 30, prob = .05),dbinom(4, size = 30, prob = .05),dbinom(5, size = 30, prob = .05),dbinom(6, size = 30, prob = .05),dbinom(7, size = 30, prob = .05),dbinom(8, size = 30, prob = .05),dbinom(9, size = 30, prob = .05),dbinom(10, size = 30, prob = .05),dbinom(11, size = 30, prob = .05),dbinom(12, size = 30, prob = .05),dbinom(13, size = 30, prob = .05),dbinom(14, size = 30, prob = .05),dbinom(15, size = 30, prob = .05),dbinom(16, size = 30, prob = .05),dbinom(17, size = 30, prob = .05),dbinom(18, size = 30, prob = .05),dbinom(19, size = 30, prob = .05),dbinom(20, size = 30, prob = .05),dbinom(21, size = 30, prob = .05),dbinom(22, size = 30, prob = .05),dbinom(23, size = 30, prob = .05),dbinom(24, size = 30, prob = .05),dbinom(25, size = 30, prob = .05),dbinom(26, size = 30, prob = .05),dbinom(27, size = 30, prob = .05),dbinom(28, size = 30, prob = .05),dbinom(29, size = 30, prob = .05),dbinom(30, size = 30, prob = .05))

plot(x,y, xlab = "# of People that Died", ylab = "Probability", main = "Binomial Distribution", col = "blue")

Poisson Distribution

dpois(5,1.5)
## [1] 0.01411996
#On average there are 1.5 deaths per 30 people per 30 days, and so the probability of observing 5 deaths is 0.014. 
# Or you could say that there is a 1.4% chance we observe 5 deaths within 30 days of treatment.