Question

 

1. Find the probability the student gets between 2 and 7, exclusive, questions correct.

2. Find the probability the student gets more than 6 questions correct.

Answer 1

Here in the quiz there are 10 multiple choice questions.

Each question has 4 answer choices.

Now, we see that student takes the quiz and randomly guesses on every question and there is only one right answer.

so here probability that he gets the right answer = 1/4

so here if x is the number of right answer he has get out of 10 questions. x will follow binomial distribution where

x ~ BIN(n = 10, p = 0.25)

p(x) = ¹⁰C_x (0.25)^x(0.75)^(10-x)

Question 1

So here we are required, the probability the student gets between 2 and 7, exclusive, questions correct.

Pr(2 < = x < = 7) = Pr(x = 2) + Pr(x = 3) + Pr(x = 4) + Pr(x = 5) + Pr(x = 6) + Pr(x = 7)

= ¹⁰C₂ (0.25)²(0.75)⁸+  ¹⁰C₃ (0.25)³(0.75)⁷ +  ¹⁰C₄ (0.25)⁴(0.75)⁶ +  ¹⁰C₅ (0.25)⁵(0.75)⁵ +  ¹⁰C₆ (0.25)⁶(0.75)⁴ +  ¹⁰C₇ (0.25)⁷(0.75)³

= 0.2816 + 0.2503 + 0.1460 + 0.0584 + 0.0162 + 0.0031 = 0.7556

Question 2

Pr(x > 6) = Pr(x = 7) + Pr(x = 8) + Pr(x = 9) + Pr(x = 10)

= ¹⁰C₇ (0.25)⁷(0.75)³ +  ¹⁰C₈ (0.25)⁸(0.75)² +  ¹⁰C₉ (0.25)⁹(0.75)¹ +  ¹⁰C₁₀ (0.25)¹⁰(0.75)⁰

= 0.0031 + 0.0004 + 0.000029 + 0.000001 = 0.9965