Question

Problem 2 (10 points). Dr. Willis is teaching two sections of Calculus I, each with 25 students. Suppose that each student will earn a passing grade, independently of other students, with probability of 0.82. What is the probability that in one of Dr. Willis's classes more than 22 students will earn a passing grade and in the other one less than 23?

Answer 1

Let x be the number of students will earn a passing grades

x follows binomial distribution with n = 25 , p = 0.82 , q = 0.18

; x = 0,1,2,…n

P(x) = (*) *p* + g(n)

Let in the class I more than 22 students will earn a passing grades

P( x > 22 ) = P(23)+ P(24)+ P(25)

=
(33) * 0.82% * 0.18/25-23
+
+ 0.824 +0.1s(25-24)
+
| * 0.825 +0.18 (25-25)

= 0.1012 + 0.0384 + 0.0070

= 0.1467

Let in the class I less than 23 students will earn a passing grades.

P( x < 23 ) = P( x ≤ 22 ) = 1 - P( x > 22 ) = 1 - 0.1467 = 0.8533

Similarly probability that in class II more than 22 students will earn a passing grades = 0.1467

and probability that in class II less than 23 students will earn a passing grades = 0.8533

P( One of the classes more than 22 students will earn a passing grade and in the other one less than 23 )  

= P( Class I more than 22 students will earn a passing grades )*P( Class II less than 23 students will earn a passing grades.) + P( Class I less than 23 students will earn a passing grades.)*P( Class II more than 22 students will earn a passing grades)

= [ 0.1467*0.8533 ] + [0.8533*0.1467]

= 0.2503