Probability of an Event

Topics: Probability

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(definition)

In the Laplacian (classic) approach of probability, given $A\subset \Omega$ an event, the probability of $A$ is defined as:

$$\mathbb{P}(A):=\frac{|A|}{|\Omega |}=\frac{\text{favorable cases where A occurs}}{\text{total cases}}$$

Example

Let $\epsilon$ be “throw a die”. Then $\Omega =\{1,2,\dots ,6\}$ ($|\Omega |=6$).

Let $A$ be “obtain an even number when the die is thrown”. Then:

• $A=\{2,4,6\}$ ($|A|=3$)
• $\mathbb{P}(A)=\frac{|A|}{|\Omega |}=\frac{3}{6}=\frac{1}{2}$

Geometrically §

When trying to determine the probability of geometric events, then $|A|$ refers to the length/area/volume/etc. where $A$ occurs and $|\Omega |$ refers to the total length/area/volume/etc.

For example, let’s say we throw a die on a circle of radius $r$. Let $A$ be “the die falls closer to the circle’s centre point than to the circumference”. Then:

$$\mathbb{P}(A)=\frac{\pi {\left(\frac{r}{2}\right)}^2}{\pi r^2}=\frac{\pi \frac{r^2}{4}}{\pi r^2}=\frac{1}{4}$$