The California License Plate Game

"The California License Plate Game", or "Count to 24", is something I play while I am driving, particularly when stuck in traffic. 

As you could see, the general California license plate, showing above, is made of "nXXXnnn", where n is a number between 0 to 9 and X letter from A to Z. The game is to calculate to 24 with those 4 numbers by only using + - x /. For example. the 4 numbers above  [6 2 4 4], could arrive 24 as 2 *(6+4) + 4. Of course, there are certainly license plates that is impossible to do so, such as [6 2 5 0].

For a long time, I have been wondering what is the overall probability for a license plate to be able to count to 24 and also want to have answers for each and every possible ones. Finally this is the result of a script I ran: https://github.com/a10jliu/licenseplategame/blob/main/Results . 

Here are the results:

    0.6902

The probability of a license place could count to 24. 

That is 6902 among the total 10000 possible combinations.

And then let's look into each numbers, what is possibility to count to 24 when the license plates contains number 3? And which number is the easiest one  ?

3 2823

4 2763

8 2694

6 2667

2 2596

5 2469

9 2424

1 2336

7 2264

0 1180

So the answer is number 3, followed by number 4 and 8;  and 0 is the least likely number to help calculation to 24.

But what is the exact percentage of a number to count 24 ? If you look at the above results carefully, you would notice that by adding them altogether, it does not ends with 0. Why is that ?

The reason behind this is, although there are 10000 possibilities, and each counts 4 numbers; so each number should (and it is) appear 10000*4/10 =4000 times. But remember we only count the number once, for example, [0 0 0 0] contains 4 zeros but we would only count zero one single time.

So instead of 4000, the total combinations of one number showing up in a license plate would be 

1         (all 4 numbers are this same number, for example [0000])

4*9       (3 numbers are the same and one is others, for example [0001])

6*9*9     (2 numbers are the same and the other 2 different, ie. [0023])

4*9*9*9   (1 number is the selected one and others different, ie [0123])

Total :  3439

With this number in hand, now we could go ahead add the percentage:

3  2823  0.8208

4  2763  0.8034

8  2694  0.7833

6  2667  0.7755

2  2596  0.7548

5  2469  0.7179

9  2424  0.7048

1  2336  0.6792

7  2264  0.6583

0  1180  0.3431

Now we know, if you see a number 3 in a license plate, there are more than 82% chance you could count to 24!