The other day, I was walking down Euclid and noticed something interesting. While I was waiting for the light to turn I noticed that one of the cars speeding by had the license plate: ‘TOY • 8765’.” Thoroughly amused, I wondered out loud, “What’s the probability of that?” As it turns out, this is a question we can answer rather easily.
As you might remember from high school, to get the probability of some event and some other event happening, you multiply the probabilities—assuming the events are what is known as statistically independent. We will make use of this fact. We will assume, as is standard for these types of problems, that each character of the license plate is chosen independently of the others and that each possible value of the character, i.e. each letter for the letter part of the plate or each number for the number part of the plate, is equally likely to come up.
Thus, every letter has a (1/26) chance of appearing and each number has a 110 chance of appearing. You can think of it as if we had a bag of marbles and each one had a number on it, zero to nine. It is clear that you would have a 110 chance of drawing, for example, the number seven or in fact any one number in the bunch.
From here, calculating the probability that a plate such as the one I saw would emerge randomly is trivial. In fact, the answer is (1/26)^3 * (1/10)^4 or about one over 175 million.
Of course, this probability is pretty low because we are calculating the probability of randomly drawing that exact plate. If we only wanted to calculate the probability that “TOY” is part of the plate the calculation simplifies to (1/26)^3, or 1/17576. This probability, while small, is far greater than the one before. The moral of the story here is that math isn’t as useless as some of you may have thought. It clearly helps answer the most pressing issues of our age … or at the very least rhetorical questions I ask myself on my walk to class.