Question

4. An exam paper consists of ten multiple choice questions, each offering four choices of which only one is correct. If a candidate chooses his answers I completely at random, what is the probability that (i) he gets at least 8 questions right, (ii) the last of the ten questions is the eighth one he gets right, (ii) in six such exams, he gets at least 8 questions right in at most one exam?

Answer 1

The probability of getting 1 question right is:

$p=0.25$

This because each question has 4 choices of which only one is correct.

1)  The probability of getting at least 8 questions right of 10 questions is:

$P(X\geq 8)= P(X=8)+P(X=9)+P(X=10)$

$=\:^{10}C_8\times 0.25^8\times 0.75^2+\:^{10}C_9\times 0.25^9\times 0.75^1+\:^{10}C_{10}\times 0.25^{10}$

$= 0.000416$

2)  In this part we have to calculate the probability of getting 7 questions right out of 9 questions and 10th questions right. That will make the 10th question the 8th one right.

$P=P(X=7)\times 0.25$

  $=\:^9C_7\times 0.25^7\times0.75^2\times 0.25=0.000309$

3)  The probability of getting at least 8 questions right in 1 exam is calculated in part 1. Now the probability of getting at least 8 questions right in atmost 1 exam of 6 exams is:

$P(X\leq1)=\:^6C_1\times 0.000416^1\times(1-0.000416)^5+ \:^6C_0\times(1-0.000416)^6$

$=0.999997$