Probability in a multiple choice test

[magnifik]

Two students are taking a 10 question test. Each question has 4 different answers. How many ways can each student answer the test? What is the probability that these two students get the same answers?

I know that to get the number of ways to answer, the number of students is irrelevant.
n^k in this case is 4¹⁰

I'm stuck on the second part of the question though. Any guidance would be helpful.

 

[daveb]

I would assume that neither knows any actual answers (otherwise, that's a really dumb HW question for the teacher to ask). Look at it this way. However the first person answers the test, you only have to have the second student match them, so what is the probability a student answers the test in a specific way, such as A, C, A, D, B, B, B, D, A, C?

 

[magnifik]

i would assume that neither knows any actual answers (otherwise, that's a really dumb homework question for the teacher to ask). Look at it this way. However the first person answers the test, you only have to have the second student match them, so what is the probability a student answers the test in a specific way, such as a, c, a, d, b, b, b, d, a, c?

1/4¹⁰?

 

[daveb]

You got it!

 

[magnifik]

You got it!

I'm wondering if it's 1/4¹⁰ or (1/4¹⁰)² since the events are independent? Or is two students getting the same score considered one event?

 

[daveb]

Let's look at it via probability.

A = event that student #1 scores a particular score: P(A) = 4^-10
B = event that student #2 scores a particular score: P(B) = 4^-10

P(A and B) = P(A|B)P(B) = (4^-10)²

However, this only accounts for one of the possible ways of answering, and there are 4¹⁰ different possible ways for a student to answer, so they can match for each of these, which gives

(4^-10)² * 4¹⁰ = 4^-10