I was going through some of the earlier answers for the license plate problems and one of the comment was as follows.
Does a plate consist of 3 letters and 3 digits in any order, like
7C99XK, or is it 3 letters followed* by 3 digits? The answers will be
different.
I understand the probability of finding a combination like AAA123 will be 1/(26^3 * 10^3). But what is the probability of finding 1AA23A?
Best Answer
The number of arrangements of three letters followed by three digits is $26^3\cdot 10^3$, seen by direct application of the rule of product using the following steps:
The total number of arrangements of three letters followed by three digits is then the product of the number of options available at each step and is then $26\cdot 26\cdot 26\cdot 10\cdot 10\cdot 10=26^3\cdot 10^3$.
IF we were to assume that every license plate consists of three letters followed by three digits in that specific order and that each such arrangement is equally likely to occur, then the probability that we correctly guess what is on the license plate assuming we guess a valid string of three letters followed by three digits will be $\frac{1}{26^3\cdot 10^3}$.
If we were to guess that the license plate contains the entire work of Shakespeare and a picture of a cow jumping over the moon though, then the probability that our guess was correct will of course be zero. Similarly so if we were to guess that the license plate was any invalid string of characters, which would in this case include things like $A1B2C3$ since this is not a string of three letters followed by three digits.
If we were to instead make the assumption that every license plate consists of three letters and three digits but these characters may appear in any order and each such arrangement is equally likely to occur, then we no longer have just $26^3\cdot 10^3$ possibilities. Rather, we approach as before with rule of product but this time also include the step "Choose which spaces in the license plate are the ones occupied by letters" and slightly reword things to work for our new situation (e.g. pick which letter occupies the furthest left available space designated for letters, etc...). The total number of arrangements in this scenario would instead be $\binom{6}{3}\cdot 26^3\cdot 10^3$.
Here still, the probability that a license plate has the entire collection of poems by Robert Frost is going to be zero as that is not a valid string of characters., but the probability that a randomly generated license plate matches a valid guess will be $\frac{1}{\binom{6}{3}\cdot 26^3\cdot 10^3}$ and that includes guesses like $AAA123$.