numbers: 4 digits that match and 1 unique digit

I have 2 random sets of numbers ranging from 0 to 9, 0 never being first but duplication allowed, that each contain 4 digits that match and 1 unique digit all in different order. What is the probability of this happening? Thank you.

example:

12,547
11,425

That's a very incoherent stream-of-consciousness style problem statement. Here's what I understand you to be asking. Please reply with any necessary corrections.

You've got two lists of 116 numbers. Each number in this list is a randomly generated integer between 10000 and 99999. Given these lists, you want to know what the chances of, let's say the 37th number from the first set and the 37th number from the second set, sharing exactly four common digits, but you don't care what the fifth digit is, nor do you care about the order of the digits. You want to know the odds of this happening nine times (is more than nine okay or must it be exactly nine?).

In addition to providing any necessary corrections, please reply with any and all work you've done on this problem, even if you know it's wrong. Thank you.

I'm only asking for clarification to ensure that we're both on the same page, and working on the same problem. I can't very well provide help if I'm working a totally different problem. I think I understand now, but I do want to ask one last question to make sure. The numbers we're comparing are the nth numbers from each of the two lists, right? Or is each number from the first list being compared against each of the 116 numbers in the second list?

In any case, I'll infer from the fact that you've shown no work means you have none to show, so we're starting at the very beginning. Because the numbers are randomly generated, we can effectively treat as digit as being random. Often it's best to work on a complex problem by starting simple and building up from there. So, let's say one of the numbers is 12345. What would be chance be of the other number having a 1 as the first digit? Repeat that question for the other four digits. What does that all suggest? Where do you think you might go from here?

I start with this.....12,547 and 11,425

12,547=12,547
11,425=12,541

11,425=11,425
12,547=17,425

What I need to know are the chances of having
12,547
11,425