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 :
P
(
A.
)
{\ displaystyle P (A)}
A.
{\ displaystyle A}
P
(
A.
)
=
Number of results in which the event
A.
entry
A.
n
z
a
H
l
a
l
l
e
r
m
O
¨
G
l
i
c
H
e
n
E.
r
G
e
b
n
i
s
s
e
{\ 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
P
(
A.
)
=
|
A.
|
|
Ω
|
{\ displaystyle P (A) = {\ frac {\ left | A \ right |} {\ left | \ Omega \ right |}}}
,
if and denote the number of elements of the event or result set .
|
A.
|
{\ displaystyle | A |}
|
Ω
|
{\ displaystyle | \ Omega |}
A.
{\ displaystyle A}
Ω
{\ 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
Ω
=
{
(
i
,
j
)
∣
i
,
j
=
1
,
...
,
6th
}
{\ 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
A.
{\ displaystyle A}
P
(
A.
)
=
4th
36
=
1
9
{\ displaystyle P (A) = {\ frac {4} {36}} = {\ frac {1} {9}}}
.
See also
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 .
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