Difference between density and probability What is the difference between the density and probability?
I have tried R in which I can use both pnorm and dnorm for the normal distribution and pbinom and dbinom for the binomial distribution, etc.
I have tried reading the documentation but I don't think it's quite clear what the difference is between those two functions. I know that pnorm(x) gives me $P(X \leq x)$.
My guess is that the density is the value of $f(x)$ while the probability is $\int_\infty^x f(x) \, \mathrm{d}x$. If this is correct, I don't understand what the value of $f(x)$ gives me of information since only the area under the function gives me the probability.
I know that the interesting values are pdf=probability density function and cdf=cumulative density function. So maybe the pdf is the value from dnorm and is the area at a specific x while cdf is the value from pnorm and is the area from $\infty$ to the specific x. But I have learned that you cannot get the probability at a specific $x$ in continuous distributions so that explanation does not make sense.
 A: ?dnorm is the density function for the normal distribution.  If you enter a quantile (i.e., a value for X), and the mean and standard deviation of the normal distribution in question, it will output the probability density.  
?pnorm is the distribution function for the normal distribution.  If you enter a quantile, and the mean and standard deviation of the normal distirubiton in question, it will output the probability of drawing a random variate from that distribution less than that quantile (you also have the option to specify greater than instead).  
(I'll add more in a bit.) 
You can also use these functions to plot the PDF and CDF of the specified distribution.  

A: I think what you said was all correct. 
dnorm is density function and pnorm is distribution function. Thus, pnorm is $F(x) = P(X <= x) = \int_{-\infty}^{x}f(x)$ and dnorm is $f(x)$.
Since $f(x) = \frac{dF(x)}{dx} = \lim_{\Delta x \to 0} \frac{F(x+ \Delta x) - F(x)}{\Delta x}$, the value of $f(x)$ shows the probability that a random variable falls in the small interval. (You could treat $f(x)$ as constant in a small interval.)
A: Everything you write is correct. As far as I understand your question you are asking, you understand why pnorm(0) = 0.5, but you ask what 
> pnorm(0)
[1] 0.5

> dnorm(0)
[1] 0.3989423

does the value dnorm(0) = 0.3989423, i.e. the height of the function mean, if it cannot be a probability? Since I cannot say it better, I refer to https://math.stackexchange.com/a/23401
