What is the likelihood of two phone numbers having several numbers match?
For example, my friend has a new work number which is xx1-x234. My cell number is xxx-1234. (The integers represent which numbers match exactly, whereas the “x” numbers do not match.) We joked around about how close the numbers were…but are they really that close? How possible is it that those numbers match by pure random chance?
Not a difficult one to answer. But it does depend a lot on what you call “matching”. In the example you give, you have 7-digit phone numbers which, with 10 possible digits per position gives you 10^7 possible combinations. You have identified four digits that match yours (in a particular non-positional way.) This leaves three digits left to fill. 10^3 = 1000. That is, 1000 out of a possible 10000000 phone numbers match in that particular way. The odds of that particular kind of matching are 1 in 10000 or 0.01%.
This is of course a slight oversimplification. Suppose you have 100 friends/acquaintances/colleagues whom you contact by phone. This gives you 100 times as much chance of getting this particular kind of matching. Now the odds appear to be 1%
Now, you and I know that the pecular kind of match you have noticed is not the only one possible. Your friend has four of the same digits in three of the same positions. It is likely that you would have noted some other kinds of similarities if they had existed. How much similarity is needed before your brain notices some kind of a match? Well, there are literally dozens of possibilities. And for each possibility you increase the odds that someone you know has got a phone number that matches yours in some way.
In other words, what at first observation seems to be a remarkable coincidence is in actual fat reasonably likely. It is unremarkable that someone you know has a phone number that has similar digital and positional arrangement to yours.
In order for the question to have a definite answer, you have to specify exactly what kind of a match you’re considering. (For example, would x1x-x234 count as “matching” xxx-1234? Would xxx-2345?)
Any particular coincidence may be highly unlikely, but the chances of some coincidence occuring ar pretty good.
By including matching digits in nearby positions, as you appear to have done with the 1, you make the probability calculation much harder. Another complication is that not all phone numbers may be valid or equally likely.
Ignoring those complications, you calculate things such as the probability that another number will match in exactly three positions (no more, no less): there are 7 x 6 x 5 / 3! = 35 groups of three digit positions in which the matching digits can lie. And there are 9 x 9 x 9 x 9 possibilities for the remaining non-matching digits. So that’s 35 x 9 x 9 x 9 x 9 = 229,635 numbers (all different, I hope, otherwise this calculation is wrong 
) that match in exactly three places, out of 10,000,000 possible numbers, for a probability of about 2.3%.
Do a similar calculation for matching in exactly 4, 5, 6 and 7 places and you could add up the probability that the other number matches in at least 3 places.
Just so I’m clear on this (statistics is not my strong suit, obviously) your 2.3% result includes ALL possible combinations of 3 consecutive numbers matching, for instance 123-xxxx, x12-3xxx, xx1-23xx, etc.?
Calculating that probability should be rather complicated. It depends situationally base off the exchange and the area code. The probability that the person living next to you has the last 4 digits in common should be rather low, since there is a higher probability that the first 6 are similar (area code and prefix). A person living on the opposite coast could have a higher probability of that last four digits because they (more than likely) won’t have the same first 6 digits. Just wanted to throw that in. Maybe it would be easier to calculate a country A vs a country B where the country code is different, but the 7 (or 10 if the OP chooses) were the same?
What would the first 6 digits have to do with the last 4?
If the first 6 digits are exactly the same, the last 4 won’t be. Unless you share a phone with them.
Someone of Asian descent has my phone number in the neighboring area code, I live in the 253, they live in the 206 area code. We use to get lots of call for them when others in the 253 area code would call them and not dial 206 first. It is irritating when the phone rings and you can’t understand what they are saying and they don’t understand you.
“You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won’t believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!”
— Richard Feynman
That’s 1 possibility in 10^6, that’s hardly enough to make a large point about them.
I once needed some legal advice. The law firm I went to had a phone number that was 1 digit away from my own. That is the same area code, and numbers abcded and abcdeb respectively.
Totally meaningless coincidence. Just thought I’d share.
What are the odds against two random people having numbers that match that closely?
For the OP’s question, I don’t think every number is equally likely to occur. My number is (xxx)66x-x666. I took the number because it was so easy to remember, but the phone company lady mentioned that most customers do not want a number with “666”.
If we’re talking about the last 4 digits, it’s 10^4. What’s that word in combinatorics where 123 is the “same” as 231? It comes up when you’re doing soduku, transposition? So that takes it to 10^3. And because of the way the OP defined “matching”, that takes it down to 10^2. We’re starting to get into the realm where it should be brought up.
That must be a US thing, or at least recent. The town of Whitby has a whiole exchange code 666 and no-one ever paid much attention to it when I was there.
We were talking about the first 6.
After reading the post I asked about again, I don’t know where the confusion was. It’s fairly obvious what he was stating, I just read it differently I guess.