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 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.
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