20. Esempi

Prova: lancio di un dado

\(\Omega\)  \(\{\) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/1Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/2Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/3Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/5Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png" alt="" width="18" height="18" role="presentation"> \(\}\), numeri naturali compresi nell’intervallo \([1, \dots , 6]\).

\(\omega\)  Esito. La faccia è un generico esito, che corrisponde al numero \(4\).

\(E\)  Evento favorevole semplice. Se l’evento favorevole è un evento semplice coincide con l’esito: \(E =\)  \(= 4\).
Evento favorevole composto. È un insieme di esiti, per esempio, le facce del dado che corrispondono ad un numero pari, \(E = \{\) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/2Nero.png?time=1619021357640" alt="" width="18" height="18" role="presentation"> ,  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation">, <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \(\} = \{2, 4, 6\} \).

\(\overline{E}\)  Evento contrario (o complementare). Se \(E = \{ \) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/2Nero.png?time=1619021357640" alt="" width="18" height="18" role="presentation"> ,  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \( \} \) allora \(\overline{E} = \{ \) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/1Nero.png" alt="" width="18" height="18" role="presentation"> ,  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/3Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/5Nero.png" alt="" width="18" height="18" role="presentation"> \( \} \).

\(E_1 \cup E_2\)  Unione di eventi. Sullo spazio campionario \(\Omega\) definiamo gli eventi:
1) Numero pari  \(E_{\text{pari}} = \{\) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/2Nero.png?time=1619021357640" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation">  \( \} = \{2, 4, 6\}\).
2) Numero \((> 3) \;\; E_{(>3)} = \{ \) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation"> ,  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/5Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \( \} = \{ 4, 5, 6 \}\).
L’evento \(E = \left( E_{\text{pari}} \cup E_ {(>3)} \right)\) è l’evento definito dalla unione di \(E_{\text{pari}}\) o \(E_{(>3)}\) e corrisponde a \(E = \{ \)  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/2Nero.png?time=1619021357640" alt="" width="18" height="18" role="presentation">, <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation">  , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/5Nero.png" alt="" width="18" height="18" role="presentation">,  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \( \} = \{2, 4, 5, 6\}\).

\(E_1 \cap E_2\)  Intersezione di eventi. Sullo spazio campionario \(\Omega\) definiamo gli eventi:
1) Numero pari  \(E_{\text{pari}} = \{\) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/2Nero.png?time=1619021357640" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \( \} = \{2, 4, 6\}\).
2) Numero \((> 3) \;\; E_{(>3)} = \{ \)  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation"> ,  <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/5Nero.png" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \( \} = \{ 4, 5, 6 \}\).
L’evento \(E = \left( E_{\text{pari}} \cap E_ {(>3)} \right)\) è l’evento definito dagli esiti che che appartengono sia all’evento \(E_{\text{pari}} \) e sia all’evento \(E_{(>3)}\) e corrisponde a: \(E = \{\) <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/4Nero.png?time=1619021396848" alt="" width="18" height="18" role="presentation"> , <img src="https://lms.federica.eu/pluginfile.php/306027/mod_book/chapter/87093/6Nero.png?time=1619021418143" alt="" width="18" height="18" role="presentation"> \(\} = \{4, 6\}\).