# 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 : ${\ displaystyle P (A)}$ ${\ displaystyle A}$

${\ displaystyle P (A) = {\ frac {{\ text {number of results in which the event}} A {\ text {occurs}}} {\ mathrm {number \ all \ m {\ ddot {o} } resembled \ results}}}}$

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 ${\ displaystyle A}$

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

## 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 .