I’m struggling in high school with this stuff.
In: 0
Science is the best way we have of studying and understanding things.
Probability is a way of filling in gaps in our knowledge in a quantitative (numbers we can work with) way.
Let’s say you need to go out tomorrow and want to know if if it is going to rain.
We can’t actually know if it will rain until tomorrow happens – while in theory it is predictable there are so many complicated variables that we can’t come to a definite answer. We could just give up at this point; maybe it will rain, maybe it won’t, might as well bring a coat. But maybe we go further.
We could build a model, using probability. An estimate of how likely it is to rain. We put some numbers into our model, and they give us back a probability. Maybe our model says that if we “run” tomorrow 100 times, 20 times it rains, the other 80 times it doesn’t. Our model gives us a probability of 20% (or 0.2, or 1 in 5) that it will rain. Now that isn’t an exact answer – it doesn’t tell us anything absolutely (we still don’t know if it will rain, and won’t until tomorrow), but it gives us something to work with. At 20% we probably bring a coat. But maybe at 5% we don’t bother.
The model helps us understand what might happen, and plan for it. It fills in the gaps in our knowledge with solid uncertainties.
Probability builds on this. It tells us how to build these models, how to work with them, and how to get out useful and interesting numbers that can help us make decisions.
Like all models there are limits – we can always add more layers to our model, make it more complicated, add more information. And if we do that we may get a different answer. We have to choose how deep to go with our model. Probability can help us make these decisions; how good an answer do we want, how important is it if we are wrong?
At high school level you are learning the basics; a few simple models, a few useful calculations (expected values, maybe standard deviations), and a few rules for calculating probabilities and combining them (tree diagrams, conditional probabilities and so on). The more you dig into the subject the more there is to discover – more complex models, fancier tools, a better understanding of the limitation of your model.
I will start by answering **”what is randomness?”**.
When you roll a die or flip a coin, the result is random, right?
Well… technically if you take the same die twice, roll it from the same position, give it the exact same speed, the exact same rotation, and have all the air molecules be at the exact same position, and have the dice be in the exact same position (up to a single atom) in your hand, and etc, then you would get the same result twice.
The only randomness here is that there are uncontrolled factors. You can’t replicate the exact same “rolling the die”, both because you don’t know all the informations about the exact position of the dice, and because it is humanly impossible to rearrange everything exactly the same.
Randomness is a mathematical tool that we use to understand better what we can’t control. And in the case of rolling a die, experience has showed that each face of the 6-sided die is obtained with a probability of 17%.
So, **what is probability science?**.
Like all science, it is decomposed in two steps: predicting and modelling.
1. When you’re doing prediction, you are give an hypothesis like “this is a fair coin with 50/50 chance of getting head or tail”, and you are tasked with computing the probability of an event, like “getting 4 heads in a row” (the answer is 7%).
2. When you’re doing modelling, you’re given a real situation with a bunch of statistics, like “we flipped this coin 153 times and obtains 82 heads and 71 tails”, and you’re tasked with finding an hypothesis like “this is a fair coin with a 50/50 chance of getting head or tail” that reasonably matches the reality.
In high school, I don’t think you will be doing the (2) very often. You’ll focus on (1), that is situations where other peoples before you already found reasonable hypothesis… or with totally artificial scenarios that have no basis in reality but give nice mathematical exercises.
Latest Answers