Two 666 licence plates

Question

I was driving to pick my girls up from school, and I noticed I was driving behind a vehicle with a licence plate number starting in 666. Then on my way back, about 30 minutes later, in the same area, but going in the opposite direction....... I realized that the car in front of me too had a licence plate starting in 666. Different cars. I live in Portland Oregon. What are the odds of this?

Answer 1

I'd like to answer how the probability depends on a few factors: (1) How license plate numbers are chosen. (2) Which license plates you see while driving. (3) How many license plates you see while driving. (4) What you are looking for.

1. How license plate numbers are chosen. If a license plate consists of three digits followed by three letters, and the digits and letters are just chosen at random, then the odds of two newly-created license plates starting with matching three digits is 1/1000. (or 1/1,000,000 if we're looking for the sequence 666 in particular.)

   (There's a 10% chance that the first digits are the same, a 10% chance that the second digits are the same, and a 10% chance that the third digits are the same, which is overall 1/1000.)

   But this assumes that license plate ids are chosen completely ("uniformly") at random. If instead license plates are generated systematically, then there may be rules for which combinations of letters and numbers are allowed, and the process for making plates might make batches that start with similar digits.

2. Which license plates you see while driving. Once the license plates are made, then it's a question of which ones you see while driving. For example, you are probably more likely to see license plates belonging to your neighbors than to people who live across town, and more likely to see plates belonging to people who travel your same routes than to people who travel on different routes; this makes it more likely to see certain samples of license plates in the same area. In the extreme case, you might just see the same car twice on separate trips in the same day, in which case you'll see a pair of license plates with the same number.

3. How many license plates you see while driving. If you're driving down the street in a sparely-populated town, the odds of you seeing similar license plates are rather low because you won't be able to check that many cars. In contrast, if you're taking a long trip down a busy city street, your chance of seeing similar license plates gets much higher. In fact, the probability of a match gets much higher than most people find intuitive—this is the so-called birthday paradox. (If 23 people are in a room, there's a 50% chance that two of them share a birthday. If 70 are in a room, the probability is 99.9% (!). Many people find these probabilities surprisingly high.)

4. What you're looking for. This is similar to how different poker hands are more or less likely: if you're looking for two license plates that match exactly, it's generally less likely than two license plates that match in the first three digits, which is less likely than two license plates that match the first digit, or have the same digits in a different order, etc.

   Humans are great pattern-finders. This means that if you drive long enough, you will see something that fits some kind of pattern, like license plates starting with the same three digits, or different people wearing the same clothes, and so on. Given our propensity to see patterns, the odds of finding some surprising pattern are very high, even if the probability of that pattern in isolation is as low as 1/1,000,000.