Question

An instructor gave her students 12 problems, telling them that 3 of the problems will be on a quiz and that passing the quiz requires solving all 3 of the problems.

a. Given that the instructor chooses the 3 problems at random, what is the probability for a student who knows only 10 problems to pass?

b. What are the chances to fail for a student who knows only 8 problems?

c. If passing the quiz requires solving at least 2 problems, what is the probability that a student who only knows 10 problems will pass?

Answer 1

Let N = Total number of problems = 12

n = number of probelms asked in quiz = 3.

Total number of ways to design a quiz = ¹²C₃ = 220

M = Number of problems student knows

N-M = Number of probelms student doesn't know.

X : Number of problems student solved.

X follows hypergeometric distribution with probability mass function is

a) passing the quiz requires all the three problems solved correctly.

Required probability = P ( X=3)

b) M = 8 and N-M = 4

Student will fail if he solved less than 3 question correctly

Required Probability = P(X=0) + P(X=1) + P(X=2)

Required probability = 0.01818 + 0.2181 +0.5090 = 0.7452

Chances to fail for a student who knows only 8 question is 74.52%

c) M = 10 and N-M = 2

Student will pass if he solved atleast two question correctly.

Required probability = P ( X =2) + P(X=3)

Required probability = 0.4090 + 0.5454 = 0.9544.