In the solution video, the answer is obtained by subtracting the probability of getting a multiple of 6 from the probability of getting an even integer. However, I don’t understand why the solution says the probability of getting a multiple of 6 from all integers is 1/6?

So I’m assuming they mean the entire number line till infinity.

Let us consider the first 6 integers

1,2,3,4,5,6

There are three multiples of 2. (2,4,6) → Set A

There are two multiples of 3. (3,6) → Set B

So now, we want multiples of 2 but not 3, so we should choose only 2 and 4 from set A but not 6 because 6 is a part of set B which we do not want. So we have a probability of 2/6.

Now repeat the same for the next 6 numbers. We can see that the pattern repeats for every set of 6 numbers. So we can assume for the whole integer space, that the same probability holds.

In other words,

There is one multiple of 6 in the subset of integers we chose

(6) → Set C

So you were told that probability of getting a multiple of 6 is 1/6.

You can see that the set A-B is the same as set A-C (both are just ignoring 6 from set A, it’s just two different approaches)