What is the probability that I live in 2 different houses with the same door number at different points in my life?

Question

The first house I used to live in at the door number 486. After a few years time, I lived in another house with the same door number 486 which I thought was incredibly lucky.

I was curious what is the probability that this can happen. Can someone explain what is the probability and how to calculate it? One of the challenges I was not able to wrap around is that, even if we assume door numbers start at 1, we don’t know at what number it stops for any given street.

Some additional information if it’s useful:

• I have lived in 13 different addresses in my entire life
• Both my 1st house and my 5th house has the same door number 486

I am not a mathematician, so forgive me for not trying to answer this myself. I hope you find this as interesting as I do.

Answer 1

As a rough guess, let's assume that there are $100,000$ possible house numbers that are all equally likely. If a person lives in $10$ different houses, the chance that they'll have a repeat house number is $$ 1 - \biggl( 1 - \frac1{100000} \biggr)\biggl( 1 - \frac2{100000} \biggr)\cdots \biggl( 1 - \frac9{100000} \biggr) \approx 0.00045 $$ (in other words, a $0.045$% chance, or about $1$ in $2{,}222$). The fact that house numbers are not equally likely means that the chances are actually higher than this.

We must also remember that:

• this is not the only coincidence that could have happened, so the odds of some interesting coincidence are much higher;
• there are billions of people in the world, so even $1$ in $2{,}222$ events happen millions of times.