What is the difference between probability and probability density?

[ray.deng83]

Like the title, I don't quite understand what probability density is and its difference with probability. Can someone explain a bit on this?

 

[blkqi]

Probability density refers to probably per "area", used to represent probably of an event within a certain interval from a continuous distribution of events. Consider for example an event: randomly picking .50000... from a hat of all real numbers in [0,1]. What's the probability of this event? Zero; 1 in infinity numbers. But what is the probability of picking a number x, such that x<.5000..., from [0,1]. One half, of course. That's a probability density.

So, the radial distribution of the hydrogen atom 1s orbital
$$P(r) = 4\pi r^{2}\psi_{100}$$
gives a distribution of probabilties of finding an electron at a radius r. Probability is then
$$P=\int_{a}^{b}4\pi r^{2}\psi_{100} \,dr$$
on the radial interval [a,b].

 

[ray.deng83]

Probability density refers to probably per "area", used to represent probably of an event within a certain interval from a continuous distribution of events. Consider for example an event: randomly picking .50000... from a hat of all real numbers in [0,1]. What's the probability of this event? Zero; 1 in infinity numbers. But what is the probability of picking a number x, such that x<.5000..., from [0,1]. One half, of course. That's a probability density.

So, the radial distribution of the hydrogen atom 1s orbital
$$P(r) = 4\pi r^{2}\psi_{100}$$
gives a distribution of probabilties of finding an electron at a radius r. Probability is then
$$P=\int_{a}^{b}4\pi r^{2}\psi_{100} \,dr$$
on the radial interval [a,b].

So, probability can be used to describe an event within a discrete distribution which has finite entities and probability density is for an interval in a continuous system. This is because there are infinite entities in a continuous system and any probability of finding a specific one is zero and hence we have to use probability density to describe it.

In this sense, for that example, can we say the probability density of finding a number x, such that 0.3<x<0.5 from [0,1] is (0.5-0.3)/1?

 

[blkqi]

In this sense, for that example, can we say the probability density of finding a number x, such that 0.3<x<0.5 from [0,1] is (0.5-0.3)/1?

Yes, because the distribution of decimals in [0,1] is uniform. In quantum mechanics we commonly deal with non-uniform wavefunctions.

 

[ray.deng83]

Yes and that's why we need those distribution functions. Thanks for your help!