Binomial distribution examples

Q1. If the likelihood of a tagged order form is 0.1, what is the probability that there are three tagged order forms in the sample of four?
solution :

k <- 3
n <- 4
p <- 0.1
dbinom(k,n,p)
## [1] 0.0036


Q2. If the likelihood of a tagged order form is 0.1, what is the probability that there are three or more (i.e., at least three) tagged order forms in the sample of four?
Solution :

\(\normalsize \sf P(X\ge3) = P(X=3)+P(X=4)\)

# P(X=3)
k = 3
n = 4
p = 0.1
Prob1 <- dbinom(k,n,p)

# P(X=4)
k = 4
n = 4
p = 0.1
Prob2 <- dbinom(k,n,p)

Prob1 + Prob2
## [1] 0.0037

Therefore, there is 0.37% of chance.

Poisson distribution examples

Q1. The number of work-related injuries per month in a manufacturing plant is known to follow a Poisson distribution, with a mean of 2.5 work-related injuries a month. Solution :

# What is the probability that in a given month, no work-related injuries occur?
mu <- 2.5
x <- 0
ppois(x,mu)
## [1] 0.082085
# That at least one work-related injury occurs?
# 1 - P(X=0)
x <- 0
1 - ppois(x,mu)
## [1] 0.917915