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Yesterday I was talking to a girl and asked her for her phone number. Once she gave it to me we realized that we got exact same last four digits. Hence out of a 8 digit phone number the last 4 digits were same. She commented that maybe it’s a strong connection and then asked “What is the probability of that happening?”

Now, I haven’t done any maths (let alone probability) in a long time.

Can someone help me understand?

I think its $\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} $, as each of the last 4 digits are unique. This translates to a $0.01\%$ chance.

Is this correct?

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    $\begingroup$ That probability assumes that all 8 digit phone numbers are accounted for in the sample space. If you could guarantee that all numbers, from 00000000 to 99999999 are in use, then yes, your probability would be .01% since your sample space would include 100,000,000 and the numbers of the form $abcdxxxx$ are 10,000. $\endgroup$ – Eleven-Eleven Apr 17 '14 at 1:50
  • $\begingroup$ @Eleven-Eleven thanks :-) I get your explanation . maybe you should have this as an answer :) $\endgroup$ – rockstar Apr 17 '14 at 1:52
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This is correct, and while it’s pretty cool, it may not be quite as significant as you think. Rare coincidences happen all the time.

Imagine I met someone new, and after we exchanged phone numbers, we noticed that they had the same first four digits. Wow! There’s only a 0.01% chance of that. What if our phone numbers had ended in $5463$ and $3645$, respectively? Wow! Those are the reverse of each other. That’s a rare coincidence. What if they were $1087$ and $9812$, which differ by exactly $8888$? Wow! That’s amazing! Maybe the eight digits of my number add up to 37, and that’s what his digits add up to. Not quite as rare, but still unusual.

If you stare at two numbers long enough, you can probably find something unusual about the pair of numbers. Wow, your phone number ends in 8734. That makes you a rare person, because only one in ten-thousand people have phone numbers ending in 8734!

The problem with getting crazy about a coincidence is that you might not be taking into account all the other coincidences that might have occurred, but didn’t. Because you noticed the specific kind of coincidence after the fact. On the other hand, if you had dreamed for years about meeting a girl with the same last four digits as you, and then you met her, well that would be really incredible, as opposed to just pretty incredible.

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