The Mysteries of Probability
The study of probability is considered by many young math students to be one of the most difficult branches of mathematics. In high school in Ontario, the only course where it taught at all is the grade 12 Data Management course, which is not specifically required by any universities or colleges, and so is the least-taken university-steam math course in that grade, a distant third behind Advanced Functions (essentially pre-calculus) and Calculus & Vectors.
What this means is that a vast majority of students never study probability in high school, and most likely never study it in a college or university either. Which is a shame for many reasons, not the least of which is that probability is the most commonly used branch of mathematics in everyday living.
"The weather report calls for a 30% chance of rain today"
You've read or heard that many times, I imagine. And if you were wondering if you would go to the beach that day, given that information, I imagine you would decide to go. I can also imagine (because it has happened to me), that you get to the beach, set up your blankets and beach umbrella, when suddenly the wind picks up, the air chills a little and within minutes it is raining so hard you decide to pack it all up and go home.
So does that mean the weather report was wrong? Should you be angry at the meteorologists? If you think about it for a few moments you realize that the answer is no - the report was not wrong. It said there was a 30% chance of rain. It did not say "I promise it will not rain". And given that information, knowing there was a chance that it could rain, you still decided to go to the beach, because it was only a "30% chance". But really, what does that mean? If the report had said there was a 90% chance, you would have stayed home. And then I can imagine that for the entire day it remained sunny. That would also be frustrating, but it wouldn't be wrong. So how is it that the meteorologists can be correct when it rains on a day when they said there is a 30% chance of rain, and also correct when it doesn't rain on a day when they said there is a 90% chance?
The answer seems clear in a sense. You can say "Well, they didn't guarantee anything in either case." Which is true. So then what did they do? And how did they do it? And how/why do we make any decisions based on the result?
Weather forecasting is an example of what we call Experimental Probability. There are actually three different kinds of probability, each with its own advantages and disadvantages. The other two are called Theoretical Probability and Subjective Probability. In experimental probability, the probability values are calculated by analyzing results from previous probability experiments, which is a way of saying we look at what has happened in the past, and calculate probabilities based on those events. For example, imagine you bought a die in a novelty store that claimed "surprising results when rolled". Like a normal die, it is a cube with the numbers 1 through 6 on each of its faces. But the "surprising results" lead you think that maybe this die favours some numbers over others, so you roll it 60 times (which is to say, you perform 60 probability experiments), with the following results:
Based on this you can informally conclude that this is not a fair die at all, and that it heavily favours a result of 6. To be precise, using experimental probability, we would say that on this die, the probability of rolling a 6 is 45/60, which is 75%.
Now, if you were to roll this die again, would you expect a 6? Would you be surprised to get a 5? The answers are probably yes and yes. But nothing is guaranteed, right? So if you rolled a 5 you wouldn't throw all your work out of the window. You would just say "Wow, that hardly ever happens," and carry on. Because the power of probability is not really in one-time events, but in the aggregate. So on this die, if you were to create a game where you bet on the number that will turn up on a roll, if you bet on 6 every time, you would come out ahead in the long run.
Weather forecasting is the same. Meteorologists analyze conditions that are known to have an impact on weather. They then compare those conditions to historical data, and look at how often rain occurred under those conditions in the past. So when they say "there is a 30% chance of rain", what they are really saying is "in the past, whenever conditions matched what we are seeing at the moment, it rained at some point during the day in 30% of cases. Now do what you must." And we do. So it might rain at the beach, but if you went to the beach every time they said "30% chance of rain", you would discover that in 70% of those days, you got to enjoy a beautiful day of sun and surf.
Theoretical and Subjective Probability
While experimental probability requires observations of past events to make probability calculations, theoretical and subjective probability calculations do not require this. In theoretical probability we start with a look at all possible outcomes of a probability experiment, and then use counting techniques to determine the probability of some event resulting. For example, if I roll a fair 12-sided die, there are 12 possible outcomes. If I am interested in the event of rolling a prime number, then there are 5 ways for that to happen (the numbers 2, 3, 5, 7 and 11 are the only possible prime numbers on this die). So the theoretical probability of rolling a prime number is 5/12 or just under 42%. Theoretical probability is used extensively in game design. In fact, casinos rely on it completely to guarantee that they turn a profit. Contrary to popular belief, casinos are not mad when people win big and they have to make huge payouts. They love it. Because the theoretical probabilities have all been determined, and in the aggregate, the casino can not lose. So when someone wins big money, it gives everyone else hope that they can as well, and that makes them play more. Which increases the aggregate!
Subjective probability is much different than either of the others. In subjective probability we use our own internal understanding of a situation to make predictions. For example, suppose there is a person you have been interested in asking out on a date. You have been interacting with them for some time, and have been looking for clues as to whether or not they would be interested. Finally, you say to yourself "well, I think there is a pretty good chance they will say yes" and you ask. That is subjective probability. You could put a number on it if you had to. You might say "There is an 80% chance they would say yes", which would really mean that you felt that you were not certain they will say yes, but that you were almost certain. We use subjective probability all the time, every day. You press the button to call the elevator, wait 5 minutes, and decide to take the stairs. That's experimental/subjective probability at work. Experimental because you are using past experience with elevators to inform your decision. Subjective because you didn't make a calculation, but rather used a "feeling" to decide that it wasn't worth waiting and that you should take the stairs (your heart will thank you , by the way).
Probability is well worth studying and understanding!
I hope I have given you a small glimpse into the mystery of probability. First, that unless someone states a probability as 0% or as 100% there is no implied guarantee. Second, that probabilities do not really show up too obviously in one-off events (like going to the beach today), but rather manifest in the aggregate. This means that decisions you make based on probabilities won't always appear to have been the correct choice (even though they are - not going to the beach when there is a 90% chance of rain is the correct choice, even if it ends up not raining), but that over time, which is to say in the aggregate, you will have benefitted from using probabilities to inform your decisions.
Which is really cool to think about!
Thanks for reading,