Question

A bank hires management trainees for entry level positions. Past expe-rience indicates that only 10% will still be employed at the end of one year.Assume the bank recently hired six trainee.

i) What is the probability that three of the trainees will still be employed at the end of one year?

ii)What is the probability that at least two of the trainees will still be em-ployed at the end of one year? Interpret.Provide the answer explicitly on your sheet using the formula, provide the corresponding R codes as well.

Answer 1

From past experience we have the knowledge that only 10% of the people employed will still remain so at the end of one year.The company hires 6 people. The probability of each of them to remain employed at the end of one year is 0.1 (we get this probability from the given percentage). The event that each of them remains employed at the end of one year is independent of the others (because the fact of someone working there does not depend on what others do or don't (hypothetically!)). Take one of the employees as an example . He/She will either remain employed at the end of 1 year with a probability of 0.1 or he/she will not with a probability of 0.9! This fact remains identical for the others as well and does not depend on the fact of what others do. So each of these employees can be thought of as a Bernoulli random variable. If X be the number of people among these 6 employees who remain employed at the end of one year,then X follows a Binomial distribution with parameters n=6 and probability of success ,p=0.1.

i) The probability that three of the 6 trainees will still be employed at the end of one year =P(X=3)==0.01458(we calculate this using R 3.6.1)

ii)The probability that atleast two of the trainees will still be employed at the end of one year is =P(X>=2)=1-P(X<2)=1- P(X=0)-P(X=1)=0.114265 (also calculated using R 3.6.1)

The R code used for the above calculations is as follows:

#to calculate probabilities of a Bin(6,0.1) distribution
#pi:probability that i employees still work at the end of one year

factorial <- function(num){
 if(num<0){
  print("Invalid input!")}
 else if(num==0){
  result=1
  return(result)}
 else{
  result=1
  for(i in 1:num){
   result=result*i}
  return(result)}
}
p3=(factorial(6)/(factorial(3)*factorial(6-3)))*(0.1^3)*(0.9^3)
p3
#[1] 0.01458
p0=(factorial(6)/(factorial(0)*factorial(6-0)))*(0.1^0)*(0.9^6)
p0
# [1] 0.531441
p1=(factorial(6)/(factorial(1)*factorial(6-1)))*(0.1^1)*(0.9^5)
p1
#[1] 0.354294
Prob_atleast2=1-p0-p1
Prob_atleast2
#[1] 0.114265

Interpreatation: Note that in terms of events the event that atleast 2 employees remain at the company at the end of one year(say,A) contains the event that 3 employees still remain at the end of one year (say B).

So by the monotonicity property of probability, ,

that is,P(event that atleast 2 employees remain at the company at the end of one year )> P(event that 3 employees still remain at the end of one year).