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
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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