Suppose a graduate program has sent acceptance letters to 45 applicants, but only has enough funding for 30 students. Let the students who were accepted to the program be independent of one another, and the chance that a student will join the program be 0.7. What is the probability that the graduate program will have enough funding for all student that join the program?

the probability that the graduate program will have enough funding for all student that join the program is 0.3653 (36.53%)

Step-by-step explanation:

since each student is independent on others the random variable X = x students of 45 applicants will join the program has a binomial probability distribution

P (X=x) = n!/[ (n-x) !*x!]*p^x * (1-p) ^x

where

n = total number of students = 45

p = probability that a student join the program = 0.7

x = number of students that join the program

then in order to have enough funding x should not surpass 30 students, then

P (X≤30) = ∑P (X) for x from 1 to 30 = F (30)

where F (30) is the cumulative probability distribution

then from binomial probability tables

P (X≤30) = F (30) = 0.3653 (36.53%)

therefore the probability that the graduate program will have enough funding for all student that join the program is 0.3653 (36.53%)