I have started to study continuous probability and probability density functions. I understand very basic concepts like the probability of an object twisted on a wheel of fortune to form an angle between 240° (excluded) and 260° (included), which turns out to be:

$$P(X \text{ angle to be between 240° and 260°}) = \frac{260 - 240}{360} = \frac{20}{360} = \frac{1}{18} = 0.0\overline{5}$$

What I do not understand is why the probability of $X$ being between the interval is expressed as an area (integral) when it comes to probability density functions.

Is it because if $f(x)$ gives a higher value (which should mean higher frequency, a data which is more frequent) when $X = x$, then the probability of X being within an infinitesimal interval containing $x$ is higher because the frequency $f(x)$ itself is higher?

In the above example, that would mean (to me) to express the previous probability indicatively as:

$$\int_{260}^{360} f(x)dx$$

Where $f(x) = \frac{1}{360}$ for any $x$ between $0$ and $360$ and $dx = 20$. However, this does not make much sense to me yet.

Could someone make a demonstration of why do we use integrals and how they relate to probability?

Thanks for the attention and sorry for my poor mathematical knowledge about this interesting topic.


Note that the two method results in the same number.

$$\int_{260}^{360} f(x)dx=$$

$$\int_{260}^{360} \frac {1}{360}dx =$$

$${1}{360}x|_{260}^{360}= $$

$$\frac {1}{360}(360-260) =$$

$$\frac {20}{360}= \frac {1}{12}$$

  • $\begingroup$ I know, the thing I do not understand is how to understand conceptually that the probability is the result of the integration... $\endgroup$ – user3019105 Mar 19 '18 at 8:32
  • $\begingroup$ Anyway, it's 1 over 18, not 12. $\endgroup$ – user3019105 Mar 19 '18 at 11:05

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