determining odds of an occurance

Factual Questions
1

The other day a friend of mine got a new phone from Walmart. They don’t activate it there, you get a card and do it from another phone. He asked me to do it for him, it’s an automated process, he had no number to transfer to this phone so they assigned him a random, available number.

My question is about the number he was given. His number is exactly one number off of my mother’s number. I don’t have any conspiracy theories or whatever about it, I do find it a highly unlikely occurrence though. How do I figure the odds of this happening is my question?

Here’s info I think will help someone help me, some won’t matter probably -

There’s 2 area codes for my area.

One covers 1,485,000 #'s, the other about 2800 numbers.

There’s appx 1.39 million people in the area being served by those area codes.

There’s appx 775 exchanges (first 3 #'s) in the main area code.

I didn’t see the figures for the newest area code.

Their number is off by one in the exchange, all else is the same. (don’t know if it matters, but the number that is different is off by 5 digits. So when I say off by one number, I mean that like a lottery ticket could be off by one number, and that number could be whatever).

My mom has had her number forever. It’s thru a different service provider. They don’t know each other, they’ve never spoke to or called each other (real sure this doesn’t matter).

I know there’ll be more info needed, I just don’t know what, and I sure couldn’t figure it from there if I did.

*Something I find unlikely, just not as unlikely, is that one of my other friends # shares the same suffix with both of their numbers.

2

The question we must always ask is not, what are the odds of this exact event occurring. The question is, what are the odds of this event, or of any other event which would be regarded as just as remarkable.

So, your friend’s number could have been off in any digit from your mom’s, and that would have been just as remarkable. It could have been off from some other phone number of someone else you knew, and that would have been just as remarkable. The phone number of someone else you know could have been similar to someone you knew, and that would have been just as remarkable. The phone numbers (any two of them) could have been similar in some other way, such as two digits being swapped. And that’s only coincidences involving phone numbers being similar, when there’s a whole universe of other possible coincidences out there.

What would be truly remarkable would be if there were no such coincidences in your life.

3

Yeah, speaking of phone number coincidences, I was driving last year and was startled to see that the car in front of me had my childhood best friend’s phone number (7 digits) as their license plate, except for the last digit. What are the odds of that? :wink: Perhaps less impressively, the current license plate I was assigned is Z followed by 6 digits. They are the first 6 digits of my cell phone number, except for the fourth digit (which actually is the 7th digit in my phone number).

With enough data points and ways of making matches, there’s going to be seemingly weird coincidences everywhere.

4

Why can someone not ask what the odds of a specific event occurring, which has occurred, and instead ask a question about some other event I know nothing about and just pull out of my ass?

And how am I to compare what I am asking about to something ‘just as remarkable’, when to do that I’d have to establish a scale of remarkableness? Is that scale not the odds basically?

I guess I thought there would be a mathmatical way of crunching some figures and putting a number to this.

*The fact I’ve been on this planet for 50 years, know a ton of people (over 200 contacts in my phone, only because I’m not as social as I used to be), and I’ve never had anything this close occur with a phone number is what makes me think it’s not super common. If this has occured to you or anyone you know (which why would they even mention such a mundane thing, so how would you know really) I would be truly surprised.

5

Given license plates can consist of letters and digits, and you happened to be behind that car on that day, I find that to be an odd occurance for sure. Doesn’t happen to you often I wouldn’t imagine or it wouldn’t have stood out to begin with.

thanks for sharing

6

If you have 200 contacts in your phone then how would you know if it happened before or even frequently? How many numbers do you recognize?

Think about the shared birthday question. In a room of just 23 people there’s a 50-50 chance of at least two people having the same birthday. In a room of 75 there’s a 99.9% chance of at least two people matching. It all depends on how you look at the problem. Is it likely you have a birthday twin in a group of 23 - not really. Is it possible that someone does - yes.

In your case, the chance that a random phone number is similar to your mom’s number is pretty low. The chance that a random phone number is similar to one in a collection of 200 is pretty high.

7

For a few reasons. One, I’ve got a good enough memory still that I recognize numbers are similar to other ones because I make it a point to try to remember numbers still, like everyone used to have to do in the 70’s thru at least early 90’s. I feel this helps to hold off dementia, if only a little.

*I remember my Visa #, exp date, 3 digit code. My library card number. Every address I’ve ever had. Every phone number I’ve ever had. I’m sure this is fairly common stuff for people to do though.

Two is, I don’t use Google to synch anything for me. I make a copy of my phone list from my phone and copy them to my computer as a backup. As a safeguard against it failing I’ve printed this list a few times over the years and keep it handy for me or possibly others at my house if needed. I’ve looked at this list enough and scanned down it to where if a number was close, I’d notice.

*It’s not uncommon to have several phone numbers with the same prefix. It only takes the suffix being the same to stand out really, so that’s just 4 numbers at first.
Commenting on the birthday comparison you made, that’s where I don’t understand stats at all. 365 days in a year and a random group of 23 people doesn’t seem like there would be a 50% chance of two people sharing a birthday. I would have no idea how to arrive at that conclusion.

8

If you want the math, you can get it here.

But for a high level explanation - the odds that 2 people out of 23 have April 15th as a birthday is low. But the odds that some pair out of 23 random people share a birthday is ~50%. It’s ~70% once you get to 30 people. One reason is there’s a lot of possible pairs out of 23 people, like Person 1 and Person 2. Or Person 1 and Person 3. Or Person 1 and Person 4. And so on. In fact, there’s 253 ways of picking 2 people out of 23. But that’s just 2 people. Even if those 2 don’t share a birthday, you can pair 2 people out of the remaining 19. And if they don’t, there’s still the remaining 17. And so on. The possibilities grow pretty fast.

And that’s the point. The specific probability “what are the odds my mother and I share April 15th as a birthday?” is low. But you don’t really care if it’s April 15th. If it was May 3rd, it would still seem remarkable. Or if it wasn’t your mother, but your cousin, it would still seem remarkable. So, the specific odds you asked about are fairly low. But within the class of “things I would find interesting”, it’s much less rare because the possible ways things can be interesting grows exponentially.

9

IIUC, the two arbitrary numbers were related as (608) 567-1234 and (608) 562-1234. Is that correct? We’d have to study details to be more precise, but in the scenario you laid out, the chance of this exact relationship might be about 1 in a million, I think.

By coincidence (:)) 1 in a million is the probability cited in Littlewood’s Law:

I round Littlewood’s Law up to “remarkable billion-to-one coincidences occur about once per lifetime.”

I have my own “ghost story” I’ve told here before. If the story were " … and within a minute …; then, within a minute …!" it might be a million-to-one coincidence, but both of those “within a minute” were actually “at that very second …!” making it billions-to-one odds-against.

10

As a species we are not great at putting coincidences in their proper context. There’s probably an evolutionary benefit in there somewhere.
Consider the thousands of ways that a strange coincidence could occurr, or the hundreds of ways in which a potential numerical near-miss could happen to a single doper on a given day, consider then the hundreds of dopers themselves or thousands of doper close contacts and the lengthy period of time over which such things may happen.

That it happened at all to anyone is pretty much a certainty. If you predicted the nature and scope of the event beforehand and then it happened then sure, that would be surprising. However, though you personally winning the lottery might be significant to you, the fact that someone wins the lottery is not. In effect we are all involved in thousands of ongoing, low probability lotteries every day. Confirmation bias means that we pay no attention to those myriad examples of non-coincidence but would consider it remarkable when they do occurr. Human nature to do so of course.

11

Because every specific event has extremely low odds.

For example, I just dealt out a random 5-card-stud hand, and got KD 5H AC 8D 10S. What were the odds of that? One in 2,598,960. But should I care? No, because any hand I could have dealt would have had the exact same odds.

Now, if I had said “I’m going to draw KD 5H AC 8D 10S”, and then let you shuffle the deck, and then drew those cards, that would be remarkable, and you’d be looking around to see what the trick was. In that case, the fact that I got that particular hand would be remarkable, because it’s the one hand I said I was going to draw.

12

And of course we never keep any count of all the non-coincidences…but we marvel at the occasional hit.

I was once told by someone promoting their religion that “There are just too many coincidences to be coincidence.”

My immediate reaction was to ask how she had calculated the correct number of coincidences. Do these come after all the “right number” coincidences happen, each day, week, month, quarter, or year…or are they randomly scattered? And if there were indeed an excess of coincidences, then how could she ever know which coincidences were the excess ones and thus attributable to a non-coincidence mechanism? And even if there were some such mechanism, how could she possibly know if the deity she worshiped were involved? She said she’d pray for me.

There is a very good reason that we have a word for coincidence.

13

Imagine I gave to each of a hundred dopers, plus their hundred closest contacts, a hundred “million sided die”. I asked those people each to roll those hundred die a hundred times a day and then report back here over the next year if any of them roll a “1”.

Such a report would not be remarkable, in fact we’d expect it. Such a report is analogous to the story from the OP and pretty much every other co-incidence out there.

14

Just to be clear, you’re saying that in your list of 200 phone numbers, there are no other pairs that only differ by one digit?

15

And, on top of that, it was within a half mile of where he lived, if you want to add another layer of improbability to it.

There’s just so many possible events and numbers and words, etc., in our lives that could intersect at any point that these 1-in-a-million plus events happen relatively frequently, as the aforementioned “Littlewood’s Law” (which I was unaware of before this thread) suggests.

There’s a nice video on the birthday problem here.

A simpler version which I just worked out. How many cards do you think you have to turn over from a fully shuffled deck of cards before two ranks match? (There’s 13 from A to K; we’re not counting jokers). In other words, how many cards to get a pair? And this is something you can test out empirically. The answer is, on average, five cards.

Does that match your intuition or surprise you? The same principle is at play with the birthday problem, except you’re matching out of 365 possible birthdays (excluding leap day) instead of 13 card ranks.

16

Or another one: Do you have a 20-sided die? How many do you think you need to roll before a number repeats? It’s only 6. With a 100-sided die it’s about 12. Note how this scales. With a 1000 sided die – were there such a thing --, you’d get even odds at about 37 rolls. Another way to think of it is if you have a group of 40 people write down a number between 1 and 1000, you’re more likely than not to find a match in the group. (And that’s assuming folks are being purely random. My intuition is that for most groups, certain numbers like, say, 666 or 420 or whatnot will show up more than others, thus leading to a higher-than-average hit result that two strike the same number.)

17

If I’m doing my math right (and I very well may not be), and only taking into account the last seven digits (not the area code), you would need – if phone numbers were purely randomly assigned and all digits from 000 0000 to 999 9999 were possible, which they are not – you’d need about 1200 contacts before a better-than-even chance that just the last digit is off. (I didn’t think about all the other permutations. You also have to take into account just the first digit being off, just the second digit being off, etc.) Basically, here I’m doing the equation 1 000 000!/(1 000 000^1200*(1 000 000-1200)!) The real mathematicians should check me.

I think. So that should be a very upper bound. The actual number will be south of that, of course, because the pool of possible numbers is more limited, and we’re not just looking to match specifically the first six digits of a seven-digit phone number.

18

I’m going to study up on stats. They truly interest me and, despite the fact I’m a fairly sharp guy in general, they often make no sense to me. Thanks for the link and thanks to Telemark for bringing that up.

19

The numbers were exactly like that, only it was the second digit of the prefix that was different. And the odds that you gave seem like they’re inline with what I was thinking they might be.

Link to your story? It sounds interesting.

20

I’m saying that there are no other numbers with everything in place, in order, and only varying by one digit - yes. Is this odd? There’s about 1.5 million numbers in my area. The only thing close, which I’d mentioned, is another friends number shares the same suffix as the two I’m asking about (the odds against that seem high to me as well).