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"> \(\} = \{4, 6\}\).