A multiplechoice test consists of 20 questions with possible

A multiple-choice test consists of 20 questions with possible answers of a, b, c, d. Estimate the probability that with random guessing the number of correct answers is at least 10. Use Excel to obtain more accuracy.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X < 10) = P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 20 9 ) * 0.25^9 * ( 1- 0.25 ) ^11 + ( 20 8 ) * 0.25^8 * ( 1- 0.25 ) ^12 + ( 20 7 ) * 0.25^7 * ( 1- 0.25 ) ^13 + ( 20 6 ) * 0.25^6 * ( 1- 0.25 ) ^14 + ( 20 5 ) * 0.25^5 * ( 1- 0.25 ) ^15 + ( 20 4 ) * 0.25^4 * ( 1- 0.25 ) ^16 + ( 20 3 ) * 0.25^3 * ( 1- 0.25 ) ^17 + ( 20 2 ) * 0.25^2 * ( 1- 0.25 ) ^18 + ( 20 1 ) * 0.25^1 * ( 1- 0.25 ) ^19 + ( 20 0 ) * 0.25^0 * ( 1- 0.25 ) ^20
= 0.9861

P( X > = 10 ) = 1 - P( X < 10) = 0.0139