{"id":40410,"date":"2024-01-26T18:29:00","date_gmt":"2024-01-26T17:29:00","guid":{"rendered":"https:\/\/www.valcon.it\/math\/?p=40297"},"modified":"2024-01-26T18:29:00","modified_gmt":"2024-01-26T17:29:00","slug":"es2015-pni-3-2","status":"publish","type":"post","link":"https:\/\/www.valcon.it\/pp\/es2015-pni-3-2\/","title":{"rendered":"Esame di Stato 2015 PNI – 3"},"content":{"rendered":"\n\n\n

Lanciando una moneta sei volte qual \u00e8 la probabilit\u00e0 che si ottenga testa \u201cal pi\u00f9\u201d due volte?
Qual \u00e8 la probabilit\u00e0 che si ottenga testa \u201calmeno\u201d due volte?<\/p>\n\n\n\n

\n\n\n\n

INVALSI \u2013 ESEMPIO 2 \u2013 Domanda 9<\/em><\/h3>\n\n\n\n

Qual \u00e8 la probabilit\u00e0 di ottenere esattamente 3 volte testa in 6 lanci di una moneta non truccata?<\/p>\n\n\n\n

\n\n\n\n

Medicina e Chirurgia – 2008 – 77<\/em><\/h3>\n\n\n\n

Qual \u00e8 la probabilit\u00e0 che lanciando 6 volte una moneta escano esattamente 4 teste?<\/p>\n\n\n\n

\n\n\n\n

Soluzione 1<\/h2>\n\n\n\n

Le sequenze di 6 lanci che soddisfano un certo numero di “testa” sono (prova a contarle…)<\/p>\n\n\n\n

Numero
Testa<\/th>		Sequenze<\/th>Numero
Sequenze<\/th>		Probabilit\u00e0<\/th><\/tr><\/thead>
0<\/th>CCCCCC<\/td>1<\/strong><\/td>\"\displaystyle \frac{1}{64}\"<\/td><\/tr>
1<\/th>CCCCCT CCCCTC CCCTCC CCTCCC CTCCCC TCCCCC<\/td>6<\/strong><\/td>\"\displaystyle \frac{6}{64}\"<\/td><\/tr>
2<\/th>CCCCTT CCCTCT CCCTTC CCTCCT CCTCTC CCTTCC CTCCCT CTCCTC
CTCTCC CTTCCC TCCCCT TCCCTC TCCTCC TCTCCC TTCCCC<\/td>		15<\/strong><\/td>\"\displaystyle \frac{15}{64}\"<\/td><\/tr>
3<\/th>CCCTTT CCTCTT CCTTCT CCTTTC CTCCTT CTCTCT CTCTTC CTTCCT
CTTCTC CTTTCC TCCCTT TCCTCT TCCTTC TCTCCT TCTCTC TCTTCC
TTCCCT TTCCTC TTCTCC TTTCCC<\/td>		20<\/strong><\/td>\"\displaystyle \frac{20}{64}\"<\/td><\/tr>
4<\/th>CCTTTT CTCTTT CTTCTT CTTTCT CTTTTC TCCTTT TCTCTT TCTTCT
TCTTTC TTCCTT TTCTCT TTCTTC TTTCCT TTTCTC TTTTCC<\/td>		15<\/strong><\/td>\"\displaystyle \frac{15}{64}\"<\/td><\/tr>
5<\/th>CTTTTT TCTTTT TTCTTT TTTCTT TTTTCT TTTTTC<\/td>6<\/strong><\/td>\"\displaystyle \frac{6}{64}\"<\/td><\/tr>
6<\/th>TTTTTT<\/td>1<\/strong><\/td>\"\displaystyle \frac{1}{64}\"<\/td><\/tr>
 <\/td> <\/td>64<\/strong><\/td>1<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Tutte le sequenze possibili sono 64<\/strong> (2^6).<\/p>\n\n\n\n

Soluzione 2<\/h2>\n\n\n\n

Utilizza la tecnica degli anagrammi<\/a><\/strong><\/p>\n\n\n\n

Numero
Testa<\/th>		Numero
Croce<\/th>		Numero
Anagrammi<\/th>		<\/th>Probabilit\u00e0<\/th><\/tr><\/thead>
0<\/th>6<\/td>\"\displaystyle \frac{6!}{0!\ 6!}\"<\/td>1<\/strong><\/td>\"\displaystyle \frac{1}{64}\"<\/td><\/tr>
1<\/th>5<\/td>\"\displaystyle \frac{6!}{1!\ 5!}\"<\/td>6<\/strong><\/td>\"\displaystyle \frac{6}{64}\"<\/td><\/tr>
2<\/th>4<\/td>\"\displaystyle \frac{6!}{2!\ 4!}\"<\/td>15<\/strong><\/td>\"\displaystyle \frac{15}{64}\"<\/td><\/tr>
3<\/th>3<\/td>\"\displaystyle \frac{6!}{3!\ 3!}\"<\/td>20<\/strong><\/td>\"\displaystyle \frac{20}{64}\"<\/td><\/tr>
4<\/th>2<\/td>\"\displaystyle \frac{6!}{4!\ 2!}\"<\/td>15<\/strong><\/td>\"\displaystyle \frac{15}{64}\"<\/td><\/tr>
5<\/th>1<\/td>\"\displaystyle \frac{6!}{5!\ 1!}\"<\/td>6<\/strong><\/td>\"\displaystyle \frac{6}{64}\"<\/td><\/tr>
6<\/th>0<\/td>\"\displaystyle \frac{6!}{6!\ 0!}\"<\/td>1<\/strong><\/td>\"\displaystyle \frac{1}{64}\"<\/td><\/tr>
 <\/td> <\/td> <\/td>64<\/strong><\/td>1<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Soluzione 3<\/h2>\n\n\n\n

Utilizza la distribuzione binomiale<\/strong><\/p>\n\n\n\n

Numero
Testa<\/th>		Numero
Croce<\/th>		Distribuzione binomiale<\/th>Probabilit\u00e0<\/th><\/tr><\/thead>
0<\/th>6<\/td>\"\displaystyle {6 \choose 0}\left(\frac{1}{2}\right)^0\left(\frac{1}{2}\right)^6\"<\/td>\"\displaystyle \frac{1}{64}\"<\/td><\/tr>
1<\/th>5<\/td>\"\displaystyle {6 \choose 1}\left(\frac{1}{2}\right)^1\left(\frac{1}{2}\right)^5\"<\/td>\"\displaystyle \frac{6}{64}\"<\/td><\/tr>
2<\/th>4<\/td>\"\displaystyle {6 \choose 2}\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^4\"<\/td>\"\displaystyle \frac{15}{64}\"<\/td><\/tr>
3<\/th>3<\/td>\"\displaystyle {6 \choose 3}\left(\frac{1}{2}\right)^3\left(\frac{1}{2}\right)^3\"<\/td>\"\displaystyle \frac{20}{64}\"<\/td><\/tr>
4<\/th>2<\/td>\"\displaystyle {6 \choose 4}\left(\frac{1}{2}\right)^4\left(\frac{1}{2}\right)^2\"<\/td>\"\displaystyle \frac{15}{64}\"<\/td><\/tr>
5<\/th>1<\/td>\"\displaystyle {6 \choose 5}\left(\frac{1}{2}\right)^5\left(\frac{1}{2}\right)^1\"<\/td>\"\displaystyle \frac{6}{64}\"<\/td><\/tr>
6<\/th>0<\/td>\"\displaystyle {6 \choose 6}\left(\frac{1}{2}\right)^6\left(\frac{1}{2}\right)^0\"<\/td>\"\displaystyle \frac{1}{64}\"<\/td><\/tr>
 <\/td> <\/td> <\/td>1<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Conclusioni<\/h2>\n\n\n\n
p(T < 2<\/strong>)<\/td>= p(T=0<\/strong>) + p(T=1<\/strong>)<\/td>= \"\displaystyle \frac{1}{64}+\frac{6}{64}\"<\/td>= \"\displaystyle \frac{7}{64}\"<\/td> = 0,109375<\/td>= 10,9375 %<\/td><\/tr>
p(T \u2264 2<\/strong>)<\/td>= p(T=0<\/strong>) + p(T=1<\/strong>) + p(T=2<\/strong>)<\/td>= \"\displaystyle \frac{1}{64}+\frac{6}{64}+\frac{15}{64}\"<\/td>= \"\displaystyle \frac{22}{64}\"<\/td>= 0,34375<\/td>= 34,375 %<\/td><\/tr>
p(T \u2265 2<\/strong>)<\/td>= 1 – p(T < 2<\/strong>)<\/td>= \"\displaystyle 1 - \frac{7}{64}\"<\/td>= \"\displaystyle \frac{57}{64}\"<\/td>= 0,890625<\/td>= 89,0625 %<\/td><\/tr>
p(T \u2265 2<\/strong>)<\/td>= p(T=2<\/strong>) + p(T=3<\/strong>) + p(T=4<\/strong>) + p(T=5<\/strong>) + p(T=6<\/strong>)<\/td>= \"\displaystyle \frac{15}{64}+\frac{20}{64}+\frac{15}{64}+\frac{6}{64}+\frac{1}{64}\"<\/td>= \"\displaystyle \frac{57}{64}\"<\/td>= 0,890625<\/td>= 89,0625 %<\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"

Lanciando una moneta sei volte qual \u00e8 la probabilit\u00e0 che si ottenga testa \u201cal pi\u00f9\u201d due volte?Qual \u00e8 la probabilit\u00e0 che si ottenga testa \u201calmeno\u201d due volte? INVALSI \u2013 ESEMPIO 2 \u2013 Domanda 9 Qual \u00e8 la probabilit\u00e0 di ottenere esattamente 3 volte testa in 6 lanci di una moneta non truccata? Medicina e Chirurgia … Leggi tutto<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[118,116],"tags":[],"class_list":["post-40410","post","type-post","status-publish","format-standard","hentry","category-probabilita-e-statistica","category-seconda-prova-mat"],"_links":{"self":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts\/40410","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/comments?post=40410"}],"version-history":[{"count":0,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts\/40410\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/media?parent=40410"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/categories?post=40410"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/tags?post=40410"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}