I couldn't access the system so I've only just read your complete question. Nevertheless I had prepared an answer below.
OK. There are 70 combinations of selecting 4 from 8 different objects because the formula for combinations is 8*7*6*5/(1*2*3*4)=70. The set of outcomes { 1 2 3 4 5 6 7 8 } in your question, is the set of objects referred to. By using the selection of a quartet of 4 different outcomes to form a 4-digit number (with the digits in ascending order), we can write down all possible 4-digit numbers: 1234, 1235, 1236, 1237, 1238, 1345, 1346, 1347, etc. 1234 represents drawing 4 numbers from the sample space set of 1 to 8. These numbers are unique and there are 70 of them. There are 35 beginning with 1, 20 beginning with 2, and so on. In fact from 1234 to 2467 there are 49 numbers in this set of 70. Therefore, as an example, we can use the probability P(x ≤ 2467) where x is an element of this set of 70. Since we know there are 49 numbers no bigger than 2467 the probability of selecting at random a 4-digit number (made up of different digits taken from 1 to 8) is 49/70=7/10=0.7.
So we've found a P(X)=0.7 exactly by stipulating that X is defined as x ≤ 2467.
That's one set. To find a different X we start at the highest number 5678 and count backwards till we have included 49 numbers. That takes us to 1358, so X is now x ≥ 1358 and P(x ≥ 1358)=0.7. So now we have two values of X (two conditions or event requirements) that give us the required probability.
In the answer you will have seen so far, I suggested perhaps adding the digits of the quartet would work. But when I investigated, it's only possible to approximate to 0.7 (I got 46/70=0.66 approximately for X=sum of digits ≤ 19. P(sum≤19)=0.66.
I don't know whether this is what the question is asking for, but I offer my solution as above as a possible answer.
So the experiment referred to consists first of drawing 4 numbers out of 8. The numbers are then arranged in numerical order to form a 4-digit number. This gives us a sample space for a different event. There are 70 elements in this set, 49 of which are less than or equal to 2467 and 49 are greater than or equal to 1358.
Why did I pick 4 numbers and not 3 or something else? Because somehow I needed to create a probability of 7/10 and then I discovered that 70 happened to be the number of combinations of 4 out of 8. Then I realised that 49/70=7/10 so all that was needed now was to invent a situation that would give me a result of 49/70. That's how I did it!
I hope this explanation is clear.