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
Probability is the chance that one event happens.
Probability density is the infinitesemal chance that an infinitesemal event happens. That means, the event is very unlikely because its very specific.
For example the chance that the temperature measured is exactly 20.00..°C with infinte zeros after the decimal point is pretty much zero.
So giving a probability makes no sense. We give it a probability density of Chance per degree range. For example 5%/°C . If you integrate over that between certain limits you can calculate a probability again, for example 19°C-21°C has a probability of 10%. And 19.9°C-20.1°C would be 1% (for a constant probability density of 5%/°C)
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.
Probability describes things with countable outcomes.
Probability density describes things with uncountable outcomes.
Pick a card, there are 52 possible values, probability of the queen of spades is 1/52.
Pick a random time of day, there are infinite possible values (assuming you can pick the time of day down to the microsecond, nanosecond, or as low as you want). So probability of any one particular time is 0.
You say “But wait! Nobody needs to pick a random time of day with infinite precision. For most people, isn’t within a second close enough? The probability of picking a particular second is 1 / (60x60x24) = 1 / 86400. (Even if you use a computer, they only measure down to the nanosecond or so, and even the universe itself has a [precision limit on time](https://en.wikipedia.org/wiki/Planck_units#Planck_time).)”
In other words, if you pick a precision and say “I don’t care about anything below this precision,” you can turn infinite outcomes (problematic, probability of any specific outcome = 0) into finite outcomes (probability works like you’d expect again).
With probability density, you define a function f(t) so that the area under f(t) is 1. Then if you want one-hour precision you divide the time axis into 24 chunks which slices the “pie” into 24 “pieces”, not necessarily equal. (Unequal pieces / a non-flat f(t) represent something that’s more likely to happen at certain times of day.) If you want one-second precision you divide the time axis into 86400 chunks. And so on.
Basically probability density lets you describe the situation in a way that, at any time, you can pick the precision and calculate probabilities. But there’s also the possibility you can do some calculations generically, deferring the choice of a specific level of precision until later.
probability = coin flip or rolling dice
because coin will not land standing up, it will always be heads or tails
dice will stop rolling with one face up and has same chance of face down,
it will always be 1/# of faces
probability density = dart game where skill of person throwing dart matters
a pro dart thrower will be more likely to hit the bullseye than miss it
so the probability distribution is not flat, but peak at the bullseye and curve down to outer rings
and if you want to know probability of just hitting high scores
(probability of not throwing dart to outer rings)
you integrate the bullseye and inner rings of the probability distribution function
I read all the responses you’ve received thus far and they are all technically wrong.
Most of them hinge on this argument: probability describes discrete outcomes and probability density describes continuous outcomes. **This is wrong**.
Probability describes the **likelihood of an outcome** in both discrete and continuous scenarios. Probability density functions exist for continuous scenarios and probability mass functions exist for discrete scenarios.
One response told you the “chance” that a continuous outcome is observed is “pretty much zero”. Note that in this sentence “chance” IS probability. And note that it’s not “pretty much zero” it is literally zero. You *can* find the *probability* of “someone being aged 7.05 years”: it’s zero. Just because a probability of zero isn’t useful does not mean they are not *probabilities*. A probability of zero does *not* mean it cannot happen or is impossible, it actually means “almost never” in statistics. Similarly a probability of 100% doesn’t mean it’s guaranteed to happen, it actually means “almost surely”.
The best response is correct to say that probability density describes the *probability* per unit value of the random variable. Funny how probability density describes the *probability* of the random variable but we “can’t use probability”. Because of this, the probability density of a random variable can **actually be larger than 1** whereas probability must be between 0 and 1 inclusive.
Note how it makes absolutely no sense to say “probability doesn’t work in continuous situations” only to attempt to replace it with the probability density, a value that can be larger than one.
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