Question

A group project estimates that thirty percent of students will do problem A in an assignment. Assume the class has four hundred
students in total, each of these student has 0.3 probability of doing problem A. Find the probability if there are at least one
hundred and thirty five students will do problem A. Use either the normal or Poisson approximation.
Justify your choice of approximation.

Answer 1

The solution to this problem is given below

400 in Let x be a random variable denoting the number of students among students total doing the problem A. Now, the choice of students are independent from each other and the probability of a student doing problem A is 0.3 X can take any of the values O,1,2 400 Now, 1 ,, X~ Bin (400» 0.3). total since the number of students is quite large Х normal pandom variable with Ext: = 400 X 0.3 = Ro and > we can approximate mean variance Vor X 400X0.3x{1-013) = 36 s since 2019

for The are binomeal (nop) • E(x) = np var (x) np (1-P). This Thus, X~ N(120 336). probability that there at least students doing problem А P(x) 135) 135 is , P X-120 436 3 135-120 √36 X-120 6 0.0062 The - P(2125) (www.za - N(0, 1) P(25-2,5) = [": p(25-3) = 1- P(Z <=)) required value of the probability is Now, since total number of students and large the probability 0.3 small and E(X)= 120 finete, 0.0062, is is is

we Poisson could approximate x pandom since variable with parameter 120, as well, But Poisson distribution is also discrete pandom variable, the computation will be than normal approximation since normal distribution a continuous distribution. more time consuming and complex than is