Answer to Question #282529 in Statistics and Probability for shubham

Question

A multiple choice test consists of 8 questions with 

3 answers of each question. A student answers 

each question by rolling a dice and checking 

the first answer if he gets 1 or 2 then second answer 

3 or 4 and third if he gets 5 or 6.to get a distinction 

He/she must secure at least 75% correct answer. 

If there is no negative marking then 

Solve the probability that student secures distinction.

Answer 1

Solution:

$p=\dfrac13, q=\dfrac23$

The probability of securing a distinction (i.e., getting correct answer of at least 6 of the 8 questions) $=P(X\ge6)=P(X=6)+P(X=7)+P(X=8)$

$=\ ^8C_6(\dfrac13)^6(\dfrac23)^2+\ ^8C_7(\dfrac13)^7(\dfrac23)^1+\ ^8C_8(\dfrac13)^8(\dfrac23)^0 \\=(\dfrac13)^6[\ ^8C_6(\dfrac23)^2+\ ^8C_7(\dfrac13)(\dfrac23)^1+\ ^8C_8(\dfrac13)^2(\dfrac23)^0] \\=\dfrac{1}{729}[28\times \dfrac{4}{9}+8\times \dfrac{2}{9}+\dfrac{1}{9}] \\=0.0196$