Why selecting 2 numbers is different from rolling 2 dice simultaneously?

Question

I have a maths exam tomorrow and this these questions are bugging me.

Question 1: Two numbers are randomly selected from set of first 6 natural numbers, find probability that selected numbers are co-prime.

Answer 1: Favorable cases are 11, but total cases are only 6C2, which is fine because if we pick 1,6 and 6,1, they are the same thing.

Question 2: Two dices are thrown simultaneously, what is the probability that the the sum of number is 6

Answer 2: Now, in this question why do we treat (2,4), and (4,2) as 2 different cases when in the above question we don’t?

I know this kind of sounds obvious but I a really confused about this, and would really appreciate some help

Answer 1

It depends on what your basic "event" is. In your case,

• for answer $1$, you chose "selection of two numbers out of six" as the base event, and by definition, that means order does not matter.
• For answer $2$, you chose "roll of two distinguishable dice" as the base event, and as such, the order matters.

Note that:

For answer 1, you could also take $(1,6)$ and $(6,1)$ as two different events. However, in that case, there are more than $11$ favorable cases. In fact, there are now $22$ favorable cases, and the probability you end up is the same, except you had to double count some things, overall complicating the solution. Taking unordered sets was the least complicated method.

For answer 2, you could also take unordered pairs as your base events. However, in that case, the events $\{1,1\}$ and $\{1,2\}$ would no longer have equal probability, so that would just complicate solving the final problem! Taking ordered pairs was the least complicated method.

Answer 2

You said that in 1st question number of total cases is $\binom62$. That means that only pairs of different numbers are considered (always good to remember that $\binom{n}{m}$ is number of $m$-element subsets in $n$-element set).

This is different from the 2nd question, where both dice can fall identically.

I think that the 1st question is sloppy, and you must guess what the question's author means by randomly selecting numbers.