Question

# A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer....

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.)

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) * px * (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

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