Question

A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 40% of the questions correct. (Round your answer to three decimal places.)

Answer 1

Solution:
Each question has two choices . TRUE and FALSE

So , probability of getting correct answer is p = 1/2 = 0.5

There are 10 questions.

n = 10

Let X be the number of correct answers .

So , X follows Binomial(10 , 0.5)

Using binomial probability formula ,

P(X = x) = (_n C _x) * p^x * (1 - p)^{n - x} ; x = 0 ,1 , 2 , ....., n

Now ,

P(At least 40% of the questions correct)

= P(At least 40% of 10 )

= P(At least 4  questions correct)

= P(X 4)

= 1 - { P(X < 4) }

= 1 - { P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) }

= 1 - { (₁₀ C ₀) * 0.5⁰ * (1 - 0.5)^{10 - 0} + (₁₀ C ₁) * 0.5¹ * (1 - 0.5)^{10 - 1} + (₁₀ C ₂) * 0.5² * (1 - 0.5)^{10 - 2} + (₁₀ C ₃) * 0.5³ * (1 - 0.5)^{10 - 3} }

= 1 - {0.0009765625 + 0.009765625 + 0.0439453125 + 0.1171875 }

= 1 - {0.171875 }

= 0.828