The probability that a professor in the business department

The probability that a professor in the business department is full time is 30%. If we randomly select 9 professors

What is the probability that 4 of them are FULL TIME?

What is the probability that 3 or fewer of them are FULL TIME?

What is the probability that 8 of them are PART TIME?

What is the probability that 2 of them are FULL TIME?

What is the probability that 4 of them are FULL TIME?

let X be the random variable denoting the number of professor in the business department is full time out of 9 professor.

now the probability that a professor in the business department is full time is 30%=0.3

so X follows a binomial distribution with parameters=n=9 and p=0.3

so X~Bin(9,0.3)

so pmf of X is P[X=x]=⁹C_x(0.3)^x(1-0.3)^9-x x=0,1,2,3,......,9

so the probability that 4 of them are full time is P[X=4]=⁹C₄(0.3)⁴(1-0.3)^9-4=0.171532 [answer]

so the probability that 3 or fewer of them are full time=P[X<=3]

=P[X=0]+P[X=1]+P[X=2]+P[X=3]=⁹C₀(0.3)⁰(1-0.3)^9-0+⁹C₁(0.3)¹(1-0.3)^9-1+⁹C₂(0.3)²(1-0.3)^9-2+⁹C₃(0.3)³(1-0.3)^9-3

=0.040354+0.155650+0.266828+0.266828=0.729659 [answer]

so the probability that 8 of them are part time=the probability that 9-8=1 of them are full time

=P[X=1]=⁹C₁(0.3)¹(1-0.3)^9-1=0.155650 [answer]

so the probability that 2 of them are full time=P[X=2]=⁹C₂(0.3)²(1-0.3)^9-2=0.266828 [answer]

the probability that 4 of them are full time is P[X=4]=⁹C₄(0.3)⁴(1-0.3)^9-4=0.171532 [answer]