Probability and Counting Arrangements

To count possibilities, draw a slot for each position, write in how many choices that slot has, and multiply. A four-digit code where each digit can be anything from nought to nine has ten choices in every slot, so the number of codes is ten times ten times ten times ten, which is ten thousand. Multiply rather than add, because every choice in the first slot can pair with every choice in the second. Adding would give forty, which answers no question anyone asked.

Now the same four digits, but each used exactly once. Once a digit has been placed it is no longer available, so each slot has one fewer choice than the one before: four, then three, then two, then only one left. Four times three times two times one is twenty-four. Notice that the digits are the same but the answer went from ten thousand to twenty-four, and the only thing that changed was whether repeats are allowed. That is what to read for.

If multiplying feels like a trick, draw a small tree. Three slots with two choices each: the first split gives two branches, each of those splits into two, and each of those splits again. Eight endings. The tree makes it obvious why you multiply — every existing branch splits again, so the count doubles at each level rather than growing by two. Drawing the tree is only practical for small numbers, but doing it once makes the multiplying rule stick.

Chance is how many outcomes you want divided by how many are possible, so the counting always comes first. Exactly one code is correct and ten thousand are possible, so a random guess has a chance of one in ten thousand. If you know the digits but not their order, only twenty-four are possible, so the chance is one in twenty-four — knowing which digits are used makes a guess roughly four hundred times more likely to work. Count the possibilities, and the chance falls out of the count.

Often more than one outcome suits you. If three of the twelve possibilities count as a win, the chance is three in twelve, which simplifies to one in four. Count what you want on top and everything possible underneath, then simplify, because the options are usually written in the simplest form and your unsimplified answer may not appear. One in four and three in twelve are the same chance written two ways, so if your answer is not listed, try simplifying it before assuming you are wrong. Quiz and exercises now.