Question

5. In a math class consisting of 21 students, suppose that 10 of the enrolled students are rated as “very proficient" in math, according to a standardized test. (a) (4 points) Suppose 4 students in the class are randomly selected to form a group and work on a project. What is the probability that all of the students in the group are rated as very proficient in math? (b) (4 points) If 4 students in the class are randomly selected, what is the probability that the majority are very proficient in math? (C) (6 points) Suppose that among the entire student population at the school, 35% are rated as very proficient in math. If another math class of 12 students is formed (assume the students are randomly selected from the school population), what is the probability that less than 2 students in the class are proficient in math?

Answer 1

Number of ways to select r items from n, nCr = n!/(r! x (n-r)!)

Number of students very proficient in math = 10

Total number of students = 21

Number of students not very proficient in math = 21 - 10 = 11

a) P(all students in the group are rated very proficient in math) = 10C4/21C4

= 210/5985

= 0.0351

b) P(majority are very proficient in math) = P(3 or 4 are very proficient in math)

= (10C3 x 11C1 / 21C4) + 0.0351

= (120 x 11 / 5985) + 0.0351

= 0.2557

c) P(very proficient in math), p = 0.35

q = 1 - p = 0.65

n = 12

Binomial distribution: P(X) = nCx p^x q^n-x

P(less than 2 are very proficient in math) = P(0) + P(1)

= 0.65¹² + 12x0.35x0.65¹¹

= 0.0424