Laplace's formula

The Laplace formula is a mathematical formula from the elementary theory of probability . If a random experiment only has a finite number of results and they all have the same probability , then the following applies to the probability of an event : ${\displaystyle P(A)}$ ${\displaystyle A}$

${\displaystyle P(A)={\frac {{\text{Anzahl der Ergebnisse, bei denen das Ereignis }}A{\text{ eintritt}}}{\mathrm {Anzahl\ aller\ m{\ddot {o}}glichen\ Ergebnisse} }}}$

or more formal

${\displaystyle P(A)={\frac {\left|A\right|}{\left|\Omega \right|}}}$,

if and denote the number of elements of the event or result set . ${\displaystyle |A|}$ ${\displaystyle |\Omega |}$ ${\displaystyle A}$ ${\displaystyle \Omega }$

The formula is named after the French mathematician and astronomer Pierre Simon Laplace (1749–1827).

example

When throwing a dice twice, there are 36 possible outcomes for the number combinations

${\displaystyle \Omega =\{(i,j)\mid i,j=1,\dotsc ,6\}}$.

If there are four results, the total is 9, namely at (6, 3), (5, 4), (4, 5), (3, 6). The probability of the event to get the sum 9 is thus given by ${\displaystyle A}$

${\displaystyle P(A)={\frac {4}{36}}={\frac {1}{9}}}$.

See also

• equal distribution
• Principle of indifference

literature

• Ulrich Krengel: Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 .