Probability maths question please

[BlueBloodedBlue]

I'm working with a young man with SEN and an obsession with numbers and letters. As he doesn't like walking, I've made up a game which involves looking at the last 3 letters of car number plates and counting those that have the letters in alphabetical order.

According to my maths (which I admit maybe flawed), the possible 3 letter options are:

AAA. BAA. CAA
AAB. BAB CAB
AAC etc
ABA
ABB
ABC
ACA
ACB
ACC

With ABC being the only 'right' answer.

Assuming my maths is right, the chances of the 'right answer is 1 therefore in 27 - about 4%.

BUT, we've been doing this for about the last 6 months and found every time that we end up with at least 25% 'rights' and sometimes higher.

Neither of us can work out why this is and I've agreed I would 'ask the Internet' why this might be.

There isn't all the combinations in circulation. Some are removed if offensive, eg sex. There is actually a job looking at combinations for potential offensiveness and weeding them out.

Maybe also there are regional variations in the letters used so you're not seeing a representative sample. Or there's too many combinations to use and they don't get to the end of the alphabet.

Your sample size is small so a small variation in what you see skews the percentage that fit the rules of your game.

The letters in the car reg nos denote where the car was registered, they aren't randomly assigned so application of statistics is meaningless

It's the first letters which denote the area, the last 3 letters (which are the ones we are looking at) are random.

Oh yes, that'll teach me to read and watch the TV at the same time 😀

I thought I followed your logic, but now I'm not so sure. How did you come to the 1 in 27?

Logic I would use:
Chance of second letter being after first letter: 1 in 2
Chance of third letter being after second letter: 1 in 2
(Ok, I'm oversimplifying a bit, as I have ignored cases where letters are the same, but I'd call that negligible)

Multiply together gives you 1 in 4, or 25%.

But I would vouch for me being totally correct. I'll have a little further think...

Your 1 in 27 is right for the sequences you have shown, but it will also be different whether your rules are for consecutive letters or any letters in alphabetical order.
eg for consecutive letters ABC is the only valid A— registration, but for ‘in order’ then ABD, ABE, ABF etc, and ACD, ACF etc all the way to AYZ are also valid which means many more valid results

Do you get repeated letters in this part of number plates?

If not, your list boils down to
ABC
ACB
BCA
BAC
CAB
CBA

Which would suggest 1 in 6.

I'm now even more sure that the previous logic was wrong!

Maybe that's where I've gone wrong.
I wasn't using ABC to actually represent those letters but in an algebraic fashion in that A is the earliest letter, B the next and C the latest in the alphabet.

So if the actual letters on the numberplate were GYB that would be represented as BCA on my list.

Does that make sense?

@Stevenage689 you can have repeating letters

when you say alphabetical order, do you mean consecutive alphabetical order or do you mean that the second letter comes after the first and the third letter comes after the second?

one problem with your reasoning could be that the chance of a letter repeating is much less than a chance of a letter being before or after it in the sequence. Say if G is the middle letter. There are 6 options from the selection bag that would be before it, only one that would be the same as it, and loads that would be after it. So ABC is much more likely than ABB or BBC.

Yes! That's what my brain was thinking but couldn't articulate.

Which brings it close to a 1 in 6:
ABC
ACB
BAC
BCA
CAB
CBA

(Obviously still less likely than this 1 in 6, as repeats are possible).

Certain combos will not be allowed, of course, which changes things further, though I'd guess doesn't change them too significantly, as you'd think they balance out.