B How are permutations and probability related?

[libervurto]

This may already be widely taught and I could be stating the obvious here, but I noticed how closely related permutations and probability are, and this gives an intuitive way to think about permutations.

For example, take a deck of 52 cards. How many possible permutations are there for the deck? Well, we can think of this another way: given a thoroughly shuffled deck, what would be the probability of us correctly guessing each card as it is dealt onto the table? Guessing the first card would be a 1/52 chance, then we turn it over to see if we were right, then we guess the second card, now with probability 1/51, then the third is 1/50, fourth is 1/49, and so on until we come to the last few cards... 1/4, 1/3, 1/2, 1/1. To successfully guess the entire permutation we need to get all of these guesses right, so we multiply their probabilities together to get 1/52!, and since we have guessed a single permutation it should be obvious that this probability is the same as saying our guess is 1 out of 52! possible permutations.

This also works for combinations that can have multiples of the same object, for example, binary digits. How many possible numbers can be made with 5 bits? Well, what is the chance of us guessing a random 5 bit number correctly? We must guess each bit correctly, with a probability of ½ for guessing each bit (since their value is completely independent) so that's a 1/(2 x 2 x 2 x 2 x 2) = 1/2⁵ chance of guessing the number correctly; and since this number is obviously one possible 5-bit number, we again flip the fraction over to find how many numbers can be made with 5 bits = 2⁵.

Is this helpful? If it causes more confusion than good or if it is too obvious a statement to make then I don't mind this being deleted.

 

[RUber]

For many people, combinations and permutations are first encountered in probability. So you are right to think of them together. Discrete probability hinges on the understanding of possible outcomes.