Rozważmy rzut kostką do gry o sześciu oczkach. Wiemy już, że
Ω
![](data:image/png;base64,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)
. Załóżmy że doświadczenie
powtarzamy 10-krotnie, z czego 2-krotnie wypada jedno oczko i 3-krotnie wypada
pięć oczek. W takim razie częstość otrzymania ścianki z jednym oczkiem wynosi
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAaCAMAAAC0JE1TAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAGa2OgAAOgBmOjqQOmaQOma2OpDbZgAAZrb/kDoAkDo6kDpmkLb/kNv/tmYAtmZmtraQtv//25A625Bm27Zm2////7Zm/9uQ/9u2//+2///bk1kjpwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbklEQVQoU5WPWQ6AIBBDB/d9Z1Puf06LEUFjTJyEj/LaoRB9johZJk+HTuVaVt5u2tkLHgDe3e6VQzpmjAW+76efFNlr/iXf3SJBfVTKicw4QZim347/aQh7hsKJVNISBWRxBAFTIzOgWkHKbtsBO+8E1mfPBu8AAAAASUVORK5CYII=)
, a częstość otrzymania
ścianki z pięcioma oczkami wynosi
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAaCAMAAAC0JE1TAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6AGa2OgAAOgBmOjqQOmaQOma2OpDbZgAAZgA6ZjqQZma2ZpDbZrb/kDoAkDo6kNv/tmYAtmY6tmZmtraQtv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///bE77M/QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcUlEQVQoU52PyQ6AIAxEW3dxF3dE+v9fKRCInDjYpIeXaadTgFjRhFgKN6FaoZr5G786r4BiSR/4PGxwJAtBowdaEKvovZiIQf02CRbPfAeQmUlE66aBplnZ4FKDaV57KAQcaaAcXtEL9j+uw9VwG7cXI0QGPR2f5bIAAAAASUVORK5CYII=)
. Jak doskonale wiemy
rezultatem powtórzenia tego doświadczenia mogą być z goła inne liczby i tak przykładowo
pięć oczek może nie wypaść wcale. Jeżeli jednak wykonamy wystarczająco dużo
doświadczeń tego typu to zauważymy, że częstość otrzymania ścianki z jednym
oczkiem jak i z pięcioma oczkami będzie przybliżać się do wartości
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAYAAAAaCAMAAACjBt2aAAAAAXNSR0IArs4c6QAAAFFQTFRFAAAAAAAAADpmAGa2OgAAOgBmOjo6OjqQOpDbZmY6ZmaQZrb/kDo6kNv/tmYAtmZmtraQtv//25A625Bm27Zm29vb2////7Zm/9uQ/9u2///b73aWIQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAUklEQVQYV32OSQ6AMAwDJ7SUtUDZWvr/hxIuSCCBDx4lsaXAS2sZII/TBUjfGESkfrcfswZUv5H7eDTtDrFSI/d+UybrnHQk3UWjn3hmA4uVIpzr/AMSwkzMmgAAAABJRU5ErkJggg==)
. Na podstawie tego
doświadczenia możemy zauważyć, że:
I. Częstość jest liczbą należącą do przedziału
![](data:image/png;base64,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)
gdzie częstość zdarzenia pewnego wynosi 1.
II. Jeśli dwa zdarzenia się wykluczają (np. wypadła ścianka z
jednym oczkiem i wypadła ścianka z pięcioma oczkami) wówczas częstość sumy tych
zdarzeń to suma częstości tych zdarzeń (![](data:image/png;base64,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)
Definicja 1Prawdopodobieństwem określonym na skończonej przestrzeni
zdarzeń elementarnych Ω nazywamy taką funkcję P, która każdemu zdarzeniu A,
Ω, przyporządkowuje liczbę rzeczywistą P(A) w
taki sposób, że:
(A1) ![](data:image/png;base64,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)
(A2) ![](data:image/png;base64,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)
(A3)
Jeśli
i
to
Warunki A1, A2 oraz A3 to aksjomaty prawdopodobieństwa.
Para (Ω, P) to przestrzeń probabilistyczna, czyli
matematyczny opis pewnego doświadczenia losowego.
Twierdzenie 1 (własności prawdopodobieństwa)Jeśli P jest prawdopodobieństwem określonym w przestrzeni
zdarzeń elementarnych Ω oraz
Ω, to:
1) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADoAAAATCAMAAAD2xu7kAAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OgBmOjpmOmaQOma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bOAyaLgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABDUlEQVQ4T9VSXVODQAzMoUUrVFsU8OvUO8n//4vuhjtoHWo7zvhgHpiQS7K7SUT+s+lT9ZH5+9k9Q5Leb/ey4vr1ZE3nYDeA83XODfTC5UThW4+30o1vw2Ynn5tahtuWv+93zhX0tNktw4ai1WbFt3gNZt0qoYTiQUJ5QbLdxOKwBXIF5Qh6gA9AZQRQyH985tceFoy5EktSQnPtAGMgZB1r8WyTxWrDacAMyQTmD4Lrl8QvXrXStyPgkTnNpUw2M8JAjRW40j9JeEowEG22ADXFYy/Tf0DYphugMI05yQSUq0R7Tvjocnxajncu78BOQnkhPJAfTqJPGfP47RCp1OycQ9yr/fX5L+3/b2JfrBQUK+xuL5oAAAAASUVORK5CYII=)
2) Jeśli
, to ![](data:image/png;base64,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)
3) ![](data:image/png;base64,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)
4) ![](data:image/png;base64,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)
5) ![](data:image/png;base64,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)
6) Jeśli zdarzenia
Ω wykluczają się parami to ![](data:image/png;base64,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)
Przykład 1
Wiemy, że
Ω,
oraz
. Dowiedź, że zdarzenia A i
B mogą się wykluczać.
Zgodnie z twierdzeniem 1 jeśli zdarzenia A i B się
wykluczają to
. Sprawdźmy czy równość ta
może zachodzić:
![](data:image/png;base64,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)
, a więc zdarzenia A i B
mogą się wykluczać, pod warunkiem że
, ckd.
Przykład 2
Wiemy, że
Ω
Oblicz
.
W oparciu o poznane własności prawdopodobieństwa liczymy
wartość wyrażenia
:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Ostatecznie otrzymujemy, że:
![](data:image/png;base64,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)
Twierdzenie 2Jeśli P jest prawdopodobieństwem określonym w przestrzeni
zdarzeń elementarnych Ω oraz
Ω, to
![](data:image/png;base64,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)
Przykład 3
Wiemy, że
Ω
Oblicz
wiedząc że sumą zdarzeń A,B,C jest przestrzeń Ω.
Korzystamy z twierdzenia 2 i poznanych aksjomatów
prawdopodobieństwa:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
, a więc zdarzenia A,B oraz
C nie mogą zajść jednocześnie.
Zadania do zrobienia
1. Dane są zdarzenia
.
Wiadomo, że
i
. Oblicz
Odp.
![](data:image/png;base64,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)
.
2. Kostka sześcienna została wykonana z
materiału, który nie jest jednorodny. Ścianka z jednym oczkiem wypada dwa razy
częściej niż każda z pozostałych ścianek. Wykonujemy jeden rzut tą kostką.
Oblicz prawdopodobieństwo otrzymania nieparzystej liczby oczek.
Odp.
.
3. W pudełku znajdują się kartki z
różnymi numerami. Prawdopodobieństwo wylosowania kartki z numerem nie większym
niż 10 jest równe
, a prawdopodobieństwo wylosowania kartki
z numerem nie mniejszym niż 10 jest równe
. Oblicz prawdopodobieństwo wylosowania
kartki z numerem 10.
Odp.
.