I’ve read several posts and can’t put my finger on the exact difference. Is there an easier example someone could give me to grasp this concept.
In: 5
It’s to do with how we add probabilities.
Usually the first things you learn about when studying are events like flipping a coin and rolling a dice. These are easy because they’re discrete, there’s only a certain number of possible outcomes. One thing you learn early on is that the sum of all probabilities adds up to 1.0, or 100%. Add up the probability of all the numbers on a die and you get 6 * 1/6 = 1.0. 1.0 represents something that’s guaranteed to happen: you are guaranteed to get one of the 6 numbers on the die.
Where the difficulty comes in is with continuous variables. These are variables where there aren’t a countable number of outcomes. Say you want to measure someone’s precise age: this is a continuous variable because you can always get a more precise answer by adding another decimal place. 7.00000000001 years is a different value from 7.0000000000001 years. There are an infinite number of possible values someone could have for this. Therefore whatever probability we assign to each outcome, they’re always going to go to infinity when you try to add them up, no matter how small you make each individual probability.
We get around this by simply not talking about precise values when doing probability with continuous variables. Instead of defining the probability of someone having an age of 7.05 years, you define a probability density function, which you can use to calculate the probability of someone having an age within a given range. You can’t find the probability of someone being aged 7.05 years, but you can find the probability that they’re aged between 7.04 and 7.06.
If you draw this on a graph, you have the variable (in this case age) on the x-axis and the probability density on the y axis. To calculate the actual probability, you find the area under the graph between two given points, and set the area under the whole graph to be equal to 1.0. The area of an infinitely thin slice of the graph is 0, so the probability of any specific number is 0. This can be a bit counterintuitive to think about: the probability of any specific number is 0, but that doesn’t mean any of those values are impossible.
This is a bit easier to see if you have some visuals, which you can see in this video: https://www.youtube.com/watch?v=ZA4JkHKZM50
If you want to know more about how to calculate probability densities, it uses integration, which you can find plenty of threads about on this sub.
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