Question

A multiple-choice quiz has 200 questions, each with 4 possible answers of which

only 1 is correct. Then the probability that sheer guesswork yields more than 30

correct answers for the 80 of the 200 problems about which the student has no

knowledge is equal to

Answer 1

A multiple-choice quiz has 200 questions, each with 4 possible answers of which

only 1 is correct. Then the probability that sheer guesswork yields more than 30

correct answers for the 80 of the 200 problems about which the student has no

knowledge is equal to

Answer :

we can use here binomial distribution :

The probability of correct answer is 1/4 = 0.25

( that is here we have 4 possible answer so each have same probability )

n = 200 ( sample size )

Therefore x = correct answer = follow binomial distribution with n and p ie ( n = 200 , p = 0.25)

We have to find probability for  sheer guesswork yields more than 30 correct answers for the 80 of the 200 problems about which the student has no knowledge ,

ie P ( x > 30 )

here n is large so we can use approximation of binomial distribution to the normal distribution .

E (x ) = mean of x = n * p = 200 * 0.25 = 20

and V( x ) = variance of x = n * p * q   where q = 1 - p    

= 200 * 0.25 * 0.75 = 15.5

SD ( x ) = standard deviation of x = sqrt ( V (x )) = sqrt ( 15.5 ) = 3.94

we can say x follows normal distribution with mean 20 and standard deviation is equal to 3.94

now probability ,

P ( x > 30 ) we know that z = ( ( x - mean ) / standard deviation) follow standard normal distribution

= P ( z > ( 30 - 20 ) / 3.94 )

= P ( z > 10 / 3.94 )

= P ( z > 2.5381 )

now use z score table

= 1 - P ( z <  2.5381 )

= 1 - 0.9945

= 0.0055