I may be using the wrong terms in the title but I read that if something is random then each character will occur an equal amounts of times. I read this when reading about the One-Time Pad cipher, each character in a plain text is given a random shift number so in the decrypted message each character occurs an equal amount of times so you cannot crack the cipher.
Now I was thinking about this and I know teachers randomize multiple choice tests. I was wondering if a multiple choice test is randomized then each letter (like choice A B C D or E ) should occur about the same amount of times. I tested this on some 25 multiple choice tests and the idea was some what true.
Is there some sort of equation that will tell you the chance that there should be equally distributed choices (like choice A B C D or E ). Or a way to find out if x characters are choice at random how many will need to be chosen for a high chances of equal distribution. Maybe if there is a four multiple choice test with x questions or if it is a five choice multiple choice test with x questions I am really trying to say is there a way to find out if should be equally distributed or if it may be what are the chances that it will be if given x things to randomized and they are randomized n times.
I know if you 4 things to randomize but only randomize them twice it will not be equal.
If the answers to the test are truly random, you should expect each answer to occur roughly the same number of times before you start the test. If, however, you are making the test and the first $10$ questions all have answer (B) according to you, that is not something weird (although it may seem so to you).
In order to analyze this, it is better to look at relative probabilities, i.e. 'what is the probability that these answers will be the right answers relative to the probability that those answers will be right.' For every set of answers, the probability of it being true is the same and therefore, when you already found out some answers that may seem rare, it is in fact just as possible to get that specific result as any other specific random set of answers.
(If this explanation isn't clear enough, feel free to ask for some clarification about the details.)