Question

In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.

Answer 1

Let, X be the number of number of questions answered correctly among total 20 questions.

The student uses tails for true and heads for false, then probability of answering true or false is same as probability of getting head and tails.

In Bernoulli trials, p is the probability of success and q is the probability of failure.

Here success is correct answer of the question which is 1 2 .

p= 1 2

Probability that the student gives wrong answer to the question is,

q=1− 1 2 = 1 2

According to binomial distribution, the probability is calculated as,

P( X=x )= C n n−x q n−x p x

Here, n is total number of trials and x is number of successes.

The probability that student answered at least 12 questions correctly is,

P( X≥12 )=P( X=12 )+P( X=13 )+⋯+P( X=20 ) = C 20 12 ( 1 2 ) 20−12 ⋅ ( 1 2 ) 12 + C 20 13 ( 1 2 ) 20−13 ⋅ ( 1 2 ) 13 +⋯+ C 20 20 ( 1 2 ) 20−20 ⋅ ( 1 2 ) 20 = C 20 12 ( 1 2 ) 20 + C 20 13 ( 1 2 ) 20 +⋯+ C 20 20 ⋅ ( 1 2 ) 20 = ( 1 2 ) 20 [ C 20 12 + C 20 13 +⋯+ C 20 20 ]