Question

Two students take the same test which consists of 5 questions, each one with 5 answers, each one with only 1 correct answer. If the students respond the test randomly

i) What is the probability that both of the students get the same number of correct answers?

ii) Find the probability that both tests are the same (assume that each test is independent from each other)

iii) What is the expected number of correct answers for each student?

iv) What is the probability that both students pass the test if they have to get at least 3 correct answers to pass it?

Answer 1

i)

Let X and Y be the number of correct answers by both students. Then,

X ~ Bin(n = 5, p = 1/5) and Y ~ Bin(n = 5, p = 1/5)

and X and Y are independent that is P(X = x, Y = y) = P(X = x) P(Y = y)

Probability that both of the students get the same number of correct answers = P(X = x, Y = x)

= P(X = 0) * P(Y = 0 ) + P(X = 1) * P(Y = 1 ) + P(X = 2) * P(Y = 2 ) + P(X = 3) * P(Y = 3 ) + P(X = 4) * P(Y = 4 ) + P(X = 5) * P(Y = 5 )

= ( ⁵C₀  0.2⁰ * 0.8⁵ )²  + ( ⁵C₁  0.2¹ * 0.8⁴ )²  + ( ⁵C₂  0.2² * 0.8³ )²  ​​​​​​​+ ( ⁵C₃  0.2³ * 0.8² )²  ​​​​​​​+ ( ⁵C₄  0.2⁴ * 0.8¹ )²  ​​​​​​​+ ( ⁵C₅  0.2⁵ * 0.8⁰ )²  ​​​​​​​

= 0.31975188

ii)

Assuming there are only two tests, then the probability that both tests are the same

= Probability that same tests was chosen by both students = (1/2) * (1/2) = 1/4

iii)

Expected number of correct answers for each student = n * p = 5 * 0.2 = 1

iv)

Probability that both students pass the test if they have to get at least 3 correct answers to pass it

= P(X 3) * P(Y 3)

= [P(X = 3) + P(X = 4) + P(X = 5)] * [P(Y = 3) + P(Y = 4) + P(Y = 5)]

= [( ⁵C₃  0.2³ * 0.8² )  ​​​​​​​+ ( ⁵C₄  0.2⁴ * 0.8¹ )  ​​​​​​​+ ( ⁵C₅  0.2⁵ * 0.8⁰ )] *   ​ [( ⁵C₃  0.2³ * 0.8² )  ​​​​​​​+ ( ⁵C₄  0.2⁴ * 0.8¹ )  ​​​​​​​+ ( ⁵C₅  0.2⁵ * 0.8⁰ )]

= [( ⁵C₃  0.2³ * 0.8² )  ​​​​​​​+ ( ⁵C₄  0.2⁴ * 0.8¹ )  ​​​​​​​+ ( ⁵C₅  0.2⁵ * 0.8⁰ )] ²

= (0.05120 + 0.00640 + 0.00032)²

= 0.0033547264