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VeryApe
Anmeldungsdatum: 10.02.2008
Beiträge: 3273
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 VeryApe Verfasst am: 09. Mai 2024 01:41 Titel: Wahrscheinlichkeitsrechnung |
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hab gerade spasshalber mal online eine Mathe Matura probiert, ob ich die überhaupt bestehen würde, aber habe ne Frage nicht richtig.
Angenommen: Es gibt 4 Fragen, jede mit 5 Antwortmöglichkeiten, und ich dürfte pro Frage 3 Antworten auswählen, dann hätte ich 12 Antworten die ich zu  Möglichkeiten kombinieren kann, dem gegenüber stehen insgesamt  Möglichkeiten, wobei nur eine Kombination richtig wäre.
Die Wahrscheinlichkeit diese zu treffen, wäre  , dass deckt sich auch mit der Multiplikation von den Wahrscheinlichkeiten pro Frage eine richtige Antwort zu treffen. 
Jetzt ging die Frage aber wie folgt:
Ein Abschnitt besteht aus 5 Fragen, mit je 4 Antworten nach Zufallsprinzip angeordnet. Eine davon ist richtig. Man hat pro Frage eine Antwortmöglichkeit.
Der Abschnitt gilt als bestanden, wenn die Hälfte aller Fragen richtig ist.
Angenommen man würde pro Frage zufällig eine Antwort auswählen, wie hoch ist die Wahrscheinlichkeit den Abschnitt zu bestehen?
Meiner Ansicht nach muß ich 3 Antworten richtig haben. Ich kann 5 Antworten tätigen, die ich zu  dreier Antwortmöglichkeiten kombinieren kann.
Dem gegenüber stehen die Auswahlmöglichkeiten an dreier Fragenkombinationen von
Also seien die Fragen A B C D E
dann kann ich ABC, ACD, ADE, usw kombinieren.
Für jede dreier Kombination an Fragen gibts  3er Antwortkombinationen.
Also insgesamt  dreier Antwortkombination, davon sind pro dreier Fragenkombination 1 richtige dreier Antwortkombination enthalten. Also gibts 10 richtige dreier Antwortkombinationen.
Die Wahrscheinlichkeit eine Richtige zu treffen wäre 
Jetzt konnte ich aber 5 Antworten tätigen, wodurch ich 10 Möglichkeiten auf meiner Seite habe, dann müsste die Wahrscheinlichkeit
 , 15,6 Prozent sein.
Stimmt aber nicht, wo liegt der Denkfehler=?
_________________
WAS IST LOS IN EUROPA? https://www.youtube.com/watch?v=a9mduhSSC5w |
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TomS
Moderator
Anmeldungsdatum: 20.03.2009
Beiträge: 18740
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| TomS Verfasst am: 09. Mai 2024 09:10 Titel: |
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Also zusammengefasst:
Es gibt N Fragen, jeweils mit M Antworten, wovon jeweils eine richtig ist und ich jeweils eine zufällig ankreuze. Wie hoch ist die Wahrscheinlichkeit p, den Test zu bestehen, wenn dazu mindestens Z der N Fragen richtig beantwortet werden müssen?
Stimmt das so?
In deinem Fall wäre Z=3, d.h. du musst 3 oder mehr Fragen richtig beantworten.
_________________
Niels Bohr brainwashed a whole generation of theorists into thinking that the job (interpreting quantum theory) was done 50 years ago. |
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VeryApe
Anmeldungsdatum: 10.02.2008
Beiträge: 3273
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Myon
Anmeldungsdatum: 04.12.2013
Beiträge: 5972
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| Myon Verfasst am: 09. Mai 2024 09:46 Titel: |
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Ich habe es auch so verstanden. Wenn also p=1/4 die Wahrscheinlichkeit ist, eine einzelne Frage richtig zu beantworten, wäre die Wahrscheinlichkeit, mindestens Z der N Fragen richtig zu beantworten (Binomialverteilung)
=\sum_{k=Z}^N\begin{pmatrix}N\\k\end{pmatrix}p^k(1-p)^{N-k})
Für Z=3, N=5 erhalte ich P=0.1035. |
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VeryApe
Anmeldungsdatum: 10.02.2008
Beiträge: 3273
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| VeryApe Verfasst am: 11. Mai 2024 00:10 Titel: |
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@Myon  , dennoch muß sich das über dreier Kombinationen lösen lassen, den Denkfehler habe ich schon.
Section Count: 400000 - won: 41503(10.380%)
-1 ... random, 0 ... dont answer question - question 1=-1, 2=-1, 3=-1, 4=-1, 5=-1
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'33211': 389(0.100%), '33212': 401(0.100%), '33213': 368(0.090%), '33214': 411(0.100%), '33221': 404(0.100%), '33222': 382(0.100%), '33223': 435(0.110%), '33224': 412(0.100%)
'33231': 390(0.100%), '33232': 418(0.100%), '33233': 394(0.100%), '33234': 397(0.100%), '33241': 394(0.100%), '33242': 420(0.110%), '33243': 389(0.100%), '33244': 380(0.100%)
'33311': 410(0.100%), '33312': 381(0.100%), '33313': 403(0.100%), '33314': 404(0.100%), '33321': 411(0.100%), '33322': 350(0.090%), '33323': 407(0.100%), '33324': 409(0.100%)
'33331': 404(0.100%), '33332': 414(0.100%), '33333': 415(0.100%), '33334': 374(0.090%), '33341': 388(0.100%), '33342': 392(0.100%), '33343': 408(0.100%), '33344': 404(0.100%)
'33411': 380(0.100%), '33412': 396(0.100%), '33413': 388(0.100%), '33414': 405(0.100%), '33421': 360(0.090%), '33422': 377(0.090%), '33423': 411(0.100%), '33424': 414(0.100%)
'33431': 391(0.100%), '33432': 378(0.090%), '33433': 411(0.100%), '33434': 378(0.090%), '33441': 373(0.090%), '33442': 402(0.100%), '33443': 357(0.090%), '33444': 380(0.100%)
'34111': 385(0.100%), '34112': 392(0.100%), '34113': 375(0.090%), '34114': 389(0.100%), '34121': 366(0.090%), '34122': 398(0.100%), '34123': 408(0.100%), '34124': 377(0.090%)
'34131': 380(0.100%), '34132': 391(0.100%), '34133': 396(0.100%), '34134': 350(0.090%), '34141': 459(0.110%), '34142': 443(0.110%), '34143': 390(0.100%), '34144': 403(0.100%)
'34211': 402(0.100%), '34212': 367(0.090%), '34213': 353(0.090%), '34214': 394(0.100%), '34221': 384(0.100%), '34222': 379(0.090%), '34223': 397(0.100%), '34224': 396(0.100%)
'34231': 408(0.100%), '34232': 411(0.100%), '34233': 391(0.100%), '34234': 387(0.100%), '34241': 400(0.100%), '34242': 404(0.100%), '34243': 372(0.090%), '34244': 394(0.100%)
'34311': 398(0.100%), '34312': 437(0.110%), '34313': 414(0.100%), '34314': 376(0.090%), '34321': 400(0.100%), '34322': 379(0.090%), '34323': 392(0.100%), '34324': 408(0.100%)
'34331': 367(0.090%), '34332': 448(0.110%), '34333': 402(0.100%), '34334': 372(0.090%), '34341': 409(0.100%), '34342': 376(0.090%), '34343': 396(0.100%), '34344': 365(0.090%)
'34411': 413(0.100%), '34412': 406(0.100%), '34413': 396(0.100%), '34414': 369(0.090%), '34421': 381(0.100%), '34422': 422(0.110%), '34423': 373(0.090%), '34424': 418(0.100%)
'34431': 375(0.090%), '34432': 350(0.090%), '34433': 374(0.090%), '34434': 408(0.100%), '34441': 390(0.100%), '34442': 400(0.100%), '34443': 369(0.090%), '34444': 409(0.100%)
'41111': 389(0.100%), '41112': 342(0.090%), '41113': 382(0.100%), '41114': 378(0.090%), '41121': 400(0.100%), '41122': 386(0.100%), '41123': 392(0.100%), '41124': 349(0.090%)
'41131': 412(0.100%), '41132': 391(0.100%), '41133': 383(0.100%), '41134': 397(0.100%), '41141': 405(0.100%), '41142': 380(0.100%), '41143': 401(0.100%), '41144': 413(0.100%)
'41211': 398(0.100%), '41212': 370(0.090%), '41213': 398(0.100%), '41214': 378(0.090%), '41221': 359(0.090%), '41222': 381(0.100%), '41223': 409(0.100%), '41224': 374(0.090%)
'41231': 387(0.100%), '41232': 386(0.100%), '41233': 406(0.100%), '41234': 394(0.100%), '41241': 397(0.100%), '41242': 403(0.100%), '41243': 336(0.080%), '41244': 379(0.090%)
'41311': 432(0.110%), '41312': 379(0.090%), '41313': 391(0.100%), '41314': 391(0.100%), '41321': 402(0.100%), '41322': 411(0.100%), '41323': 384(0.100%), '41324': 373(0.090%)
'41331': 382(0.100%), '41332': 402(0.100%), '41333': 411(0.100%), '41334': 381(0.100%), '41341': 369(0.090%), '41342': 368(0.090%), '41343': 376(0.090%), '41344': 384(0.100%)
'41411': 394(0.100%), '41412': 387(0.100%), '41413': 376(0.090%), '41414': 379(0.090%), '41421': 336(0.080%), '41422': 395(0.100%), '41423': 408(0.100%), '41424': 400(0.100%)
'41431': 384(0.100%), '41432': 378(0.090%), '41433': 388(0.100%), '41434': 402(0.100%), '41441': 388(0.100%), '41442': 397(0.100%), '41443': 385(0.100%), '41444': 353(0.090%)
'42111': 376(0.090%), '42112': 383(0.100%), '42113': 371(0.090%), '42114': 419(0.100%), '42121': 400(0.100%), '42122': 398(0.100%), '42123': 399(0.100%), '42124': 401(0.100%)
'42131': 413(0.100%), '42132': 380(0.100%), '42133': 388(0.100%), '42134': 402(0.100%), '42141': 366(0.090%), '42142': 438(0.110%), '42143': 448(0.110%), '42144': 388(0.100%)
'42211': 361(0.090%), '42212': 395(0.100%), '42213': 425(0.110%), '42214': 385(0.100%), '42221': 392(0.100%), '42222': 370(0.090%), '42223': 413(0.100%), '42224': 398(0.100%)
'42231': 405(0.100%), '42232': 394(0.100%), '42233': 392(0.100%), '42234': 373(0.090%), '42241': 362(0.090%), '42242': 425(0.110%), '42243': 368(0.090%), '42244': 377(0.090%)
'42311': 428(0.110%), '42312': 384(0.100%), '42313': 397(0.100%), '42314': 351(0.090%), '42321': 354(0.090%), '42322': 368(0.090%), '42323': 378(0.090%), '42324': 386(0.100%)
'42331': 379(0.090%), '42332': 389(0.100%), '42333': 396(0.100%), '42334': 363(0.090%), '42341': 385(0.100%), '42342': 357(0.090%), '42343': 352(0.090%), '42344': 427(0.110%)
'42411': 398(0.100%), '42412': 359(0.090%), '42413': 379(0.090%), '42414': 392(0.100%), '42421': 424(0.110%), '42422': 405(0.100%), '42423': 389(0.100%), '42424': 418(0.100%)
'42431': 385(0.100%), '42432': 368(0.090%), '42433': 380(0.100%), '42434': 428(0.110%), '42441': 398(0.100%), '42442': 376(0.090%), '42443': 391(0.100%), '42444': 391(0.100%)
'43111': 353(0.090%), '43112': 396(0.100%), '43113': 409(0.100%), '43114': 414(0.100%), '43121': 362(0.090%), '43122': 361(0.090%), '43123': 400(0.100%), '43124': 404(0.100%)
'43131': 422(0.110%), '43132': 420(0.110%), '43133': 393(0.100%), '43134': 359(0.090%), '43141': 415(0.100%), '43142': 394(0.100%), '43143': 414(0.100%), '43144': 389(0.100%)
'43211': 377(0.090%), '43212': 402(0.100%), '43213': 404(0.100%), '43214': 376(0.090%), '43221': 391(0.100%), '43222': 343(0.090%), '43223': 397(0.100%), '43224': 394(0.100%)
'43231': 413(0.100%), '43232': 368(0.090%), '43233': 383(0.100%), '43234': 367(0.090%), '43241': 361(0.090%), '43242': 395(0.100%), '43243': 380(0.100%), '43244': 385(0.100%)
'43311': 393(0.100%), '43312': 394(0.100%), '43313': 373(0.090%), '43314': 376(0.090%), '43321': 368(0.090%), '43322': 386(0.100%), '43323': 369(0.090%), '43324': 377(0.090%)
'43331': 375(0.090%), '43332': 382(0.100%), '43333': 394(0.100%), '43334': 369(0.090%), '43341': 379(0.090%), '43342': 397(0.100%), '43343': 409(0.100%), '43344': 404(0.100%)
'43411': 377(0.090%), '43412': 385(0.100%), '43413': 354(0.090%), '43414': 401(0.100%), '43421': 369(0.090%), '43422': 410(0.100%), '43423': 391(0.100%), '43424': 369(0.090%)
'43431': 401(0.100%), '43432': 404(0.100%), '43433': 410(0.100%), '43434': 431(0.110%), '43441': 406(0.100%), '43442': 401(0.100%), '43443': 392(0.100%), '43444': 383(0.100%)
'44111': 390(0.100%), '44112': 380(0.100%), '44113': 400(0.100%), '44114': 385(0.100%), '44121': 413(0.100%), '44122': 390(0.100%), '44123': 409(0.100%), '44124': 396(0.100%)
'44131': 348(0.090%), '44132': 381(0.100%), '44133': 390(0.100%), '44134': 412(0.100%), '44141': 346(0.090%), '44142': 401(0.100%), '44143': 394(0.100%), '44144': 369(0.090%)
'44211': 373(0.090%), '44212': 437(0.110%), '44213': 398(0.100%), '44214': 403(0.100%), '44221': 410(0.100%), '44222': 404(0.100%), '44223': 410(0.100%), '44224': 377(0.090%)
'44231': 409(0.100%), '44232': 426(0.110%), '44233': 409(0.100%), '44234': 377(0.090%), '44241': 373(0.090%), '44242': 374(0.090%), '44243': 381(0.100%), '44244': 382(0.100%)
'44311': 389(0.100%), '44312': 397(0.100%), '44313': 402(0.100%), '44314': 368(0.090%), '44321': 403(0.100%), '44322': 384(0.100%), '44323': 384(0.100%), '44324': 378(0.090%)
'44331': 409(0.100%), '44332': 393(0.100%), '44333': 407(0.100%), '44334': 333(0.080%), '44341': 398(0.100%), '44342': 400(0.100%), '44343': 353(0.090%), '44344': 365(0.090%)
'44411': 379(0.090%), '44412': 416(0.100%), '44413': 363(0.090%), '44414': 378(0.090%), '44421': 397(0.100%), '44422': 416(0.100%), '44423': 364(0.090%), '44424': 415(0.100%)
'44431': 397(0.100%), '44432': 389(0.100%), '44433': 391(0.100%), '44434': 396(0.100%), '44441': 415(0.100%), '44442': 382(0.100%), '44443': 393(0.100%), '44444': 383(0.100%)
Question 1 asked 400000
Answer 1 correct: 99887 - 24.970%
Answer 2 correct: 99987 - 25.000%
Answer 3 correct: 100551 - 25.140%
Answer 4 correct: 99575 - 24.890%
Question 2 asked 400000
Answer 1 correct: 100109 - 25.030%
Answer 2 correct: 99942 - 24.990%
Answer 3 correct: 99762 - 24.940%
Answer 4 correct: 100187 - 25.050%
Question 3 asked 400000
Answer 1 correct: 100072 - 25.020%
Answer 2 correct: 99907 - 24.980%
Answer 3 correct: 100155 - 25.040%
Answer 4 correct: 99866 - 24.970%
Question 4 asked 400000
Answer 1 correct: 99724 - 24.930%
Answer 2 correct: 100340 - 25.090%
Answer 3 correct: 100210 - 25.050%
Answer 4 correct: 99726 - 24.930%
Question 5 asked 400000
Answer 1 correct: 99862 - 24.970%
Answer 2 correct: 100487 - 25.120%
Answer 3 correct: 99844 - 24.960%
Answer 4 correct: 99807 - 24.950%
| STRG+SHIFT+J: Javascript console hat Folgendes geschrieben: |
"use strict";
var questions_per_section=5, answers_per_question=4, correct_answer_for_win=3, section_count=400000, percent_comma=3, values_per_row=answers_per_question*2,
//-- answer indices for question 1 2 3 4 5
answer_indices_arr=[undefined,-1,-1,-1,-1,-1], //-- -1 means generate random answer, 0 means do not answer the question, use this answer indices set for question 1=1, 2=2, 3=1, 4=2, 5=1
i, u, question, ca_indices_str, chosen_answer_index, hit_counter;
const section={ count:0, won:0, questions:[undefined], ca_indices_counters:{} };
function Question (index, answers_per_question)
{
var i;
this.index=index;
this.count=0;
this.correct_answer_index=null;
this.ca_index_counters=[undefined];
for (i=1; i<=answers_per_question; i++) this.ca_index_counters.push(0);
}
Question.prototype.print_stat=function()
{
var i;
console.log ("Question "+this.index+" asked "+this.count);
for (i=1;i<this.ca_index_counters.length;i++) console.log ("Answer "+i+" correct: "+this.ca_index_counters[i]+" - "+(Math.round(this.ca_index_counters[i]/this.count*10000)/100).toFixed(percent_comma)+"%");
}
section.print_stat=function ()
{
var i, ca_indices_str, output="", head;
head="Section Count: "+this.count+" - won: "+this.won+"("+(Math.round(this.won/this.count*10000)/100).toFixed(percent_comma)+"%)\n-1 ... random, 0 ... dont answer question - question";
for (i=1;i<answer_indices_arr.length;i++)
{
head+=" "+i+"="+answer_indices_arr[i];
if (i==questions_per_section) head+="\n";
else head+=",";
}
console.log (head.padEnd (head.length*2-1,"=")); i=0;
for (ca_indices_str in this.ca_indices_counters)
{
output+="'"+ca_indices_str+"':"+this.ca_indices_counters[ca_indices_str].toString().padStart(6," ")+"("+(Math.round(this.ca_indices_counters[ca_indices_str]/this.count*10000)/100).toFixed(percent_comma)+"%), ";
if (++i == answers_per_question*2) { console.log (output.slice(0,-2)); i=0; output="" };
}
if (output) console.log (output.slice(0,-2));
}
for (i=1;i<=questions_per_section;i++) section.questions.push(new Question (i, answers_per_question));
for (i=0;i<section_count;i++)
{
ca_indices_str=""; hit_counter=0;
for (u=1; u<=questions_per_section; u++)
{
question=section.questions[u];
question.correct_answer_index=Math.ceil(Math.random()*answers_per_question); //-- random correct answer index for question
question.ca_index_counters[question.correct_answer_index]++;
question.count++;
ca_indices_str+=question.correct_answer_index;
if (answer_indices_arr[u]==-1) chosen_answer_index=Math.ceil(Math.random()*answers_per_question);
else chosen_answer_index=answer_indices_arr[u];
if (chosen_answer_index==question.correct_answer_index) hit_counter++;
}
if (hit_counter>=correct_answer_for_win) section.won++;
if (section.ca_indices_counters.hasOwnProperty(ca_indices_str)) section.ca_indices_counters[ca_indices_str]++;
else section.ca_indices_counters[ca_indices_str]=1;
section.count++;
}
section.print_stat();
for (i=1;i<section.questions.length;i++) section.questions[i].print_stat();
|
_________________
WAS IST LOS IN EUROPA? https://www.youtube.com/watch?v=a9mduhSSC5w |
|
|
VeryApe
Anmeldungsdatum: 10.02.2008
Beiträge: 3273
|
| VeryApe Verfasst am: 11. Mai 2024 00:56 Titel: |
|
|
Wenn ich zufällig eine dreier Kombination an Fragen wähle, die ich beantworte,
dann gibts 10 mögliche dreier Kombinationen, 1/10 wahrscheinlichkeit für eine,
1/10 der Spiele, habe ich dann eine gewisse dreier Fragen Kombination. pro dreier Kombination gibts 4^3 Antwortmöglichkeiten. 1/4^3 Wahrscheinlichkeit für eine gewisse Antwortkombination,
1/10*1/4^3 Wahrscheinlichkeits - Treffer für eine zufällige dreier Kombination an Fragen und dabei immer dieselben Antworten, das Ganze habe ich dann aber 10 mal bei einer genügend großen Durchlaufzahl, weil alle dreier Kombination vertreten sind, sodass ich insgesamt die Wahrscheinlichkeit von
 habe.
wähle ich auch noch die Antwortkombinationen zufällig, dann habe ich
in 1/10 der Spiele und davon wieder 1/4^3 der Spiele, eine gewisse Antwortkombination, aber das Ganze habe ich dann wieder bei einer genügend großen Durchlaufzahl, 4^3 mal. sodass ich wiederum die Wahrscheinlichkeit
von
 habe.
Wenn ich alle 5 Fragen und deren Antwortkombinationen betrachte, dann habe ich eine Wahrscheinlichkeit von 4^5 für eine komplette Antwortkombination.
wenn ich jetzt wieder eine dreier Fragenkombination auswähle und dort eine gewisse Antwortkombination wähle, dann habe ich 4^2 Möglichkeiten auf meiner Seite.
Es müsste also ziemlich egal sein ob, ich die dreier Kombination an Fragen und die Antworten dazu zufällig wähle, oder ob ich alles mit fixen Werten durchlaufen lasse und das ist es auch, wenn ich es austeste.
Der Fehler liegt bei den Möglichen dreier Kombinationen, wenn ich z.B zu den drei Antworten auf drei Fragen eine weitere Antwort hinzu nehme.
Wenn ich 4 Antworten tätigen darf, kann ich 4*3*2/2*3 Mögliche dreier Kombinationen bilden.
seien die Fragen A, B , C , D, E
und ich wähle ABCD,
dann geht ABC, ABD, ACD, BCD
Die erste Kombination (z.B. ABC) bringt 4^2 Möglichkeiten, jede weitere (ABD,ACD,BCD) bringt aber nur 4^2-4 Möglichkeiten hinzu, weil sie ineinander verzahnt sind, sodass ich dann eine Wahrscheinlichkeit von
} {4^5}=0.5078125) habe
ich dachte, ich hätte 
Section Count: 1000000 - won: 50601(5.060%)
-1 ... random, 0 ... dont answer question - question 1=0, 2=-1, 3=-1, 4=-1, 5=-1
============================================================================================================================
'11111': 997(0.100%), '11112': 986(0.100%), '11113': 969(0.100%), '11114': 957(0.100%), '11121': 963(0.100%), '11122': 990(0.100%), '11123': 968(0.100%), '11124': 1007(0.100%)
'11131': 930(0.090%), '11132': 1048(0.100%), '11133': 924(0.090%), '11134': 1047(0.100%), '11141': 941(0.090%), '11142': 897(0.090%), '11143': 972(0.100%), '11144': 1043(0.100%)
'11211': 947(0.090%), '11212': 979(0.100%), '11213': 1016(0.100%), '11214': 1041(0.100%), '11221': 965(0.100%), '11222': 986(0.100%), '11223': 1008(0.100%), '11224': 967(0.100%)
'11231': 977(0.100%), '11232': 955(0.100%), '11233': 978(0.100%), '11234': 996(0.100%), '11241': 962(0.100%), '11242': 937(0.090%), '11243': 980(0.100%), '11244': 919(0.090%)
'11311': 948(0.090%), '11312': 939(0.090%), '11313': 978(0.100%), '11314': 923(0.090%), '11321': 986(0.100%), '11322': 1029(0.100%), '11323': 1025(0.100%), '11324': 976(0.100%)
'11331': 971(0.100%), '11332': 931(0.090%), '11333': 1048(0.100%), '11334': 993(0.100%), '11341': 955(0.100%), '11342': 960(0.100%), '11343': 960(0.100%), '11344': 969(0.100%)
'11411': 981(0.100%), '11412': 988(0.100%), '11413': 958(0.100%), '11414': 957(0.100%), '11421': 1004(0.100%), '11422': 992(0.100%), '11423': 972(0.100%), '11424': 981(0.100%)
'11431': 986(0.100%), '11432': 957(0.100%), '11433': 965(0.100%), '11434': 973(0.100%), '11441': 996(0.100%), '11442': 955(0.100%), '11443': 911(0.090%), '11444': 1043(0.100%)
'12111': 1016(0.100%), '12112': 943(0.090%), '12113': 917(0.090%), '12114': 990(0.100%), '12121': 985(0.100%), '12122': 967(0.100%), '12123': 995(0.100%), '12124': 983(0.100%)
'12131': 1011(0.100%), '12132': 961(0.100%), '12133': 976(0.100%), '12134': 892(0.090%), '12141': 981(0.100%), '12142': 986(0.100%), '12143': 993(0.100%), '12144': 957(0.100%)
'12211': 956(0.100%), '12212': 1009(0.100%), '12213': 1028(0.100%), '12214': 949(0.090%), '12221': 964(0.100%), '12222': 1000(0.100%), '12223': 946(0.090%), '12224': 961(0.100%)
'12231': 951(0.100%), '12232': 929(0.090%), '12233': 992(0.100%), '12234': 938(0.090%), '12241': 965(0.100%), '12242': 984(0.100%), '12243': 949(0.090%), '12244': 970(0.100%)
'12311': 997(0.100%), '12312': 976(0.100%), '12313': 989(0.100%), '12314': 965(0.100%), '12321': 952(0.100%), '12322': 957(0.100%), '12323': 992(0.100%), '12324': 996(0.100%)
'12331': 920(0.090%), '12332': 934(0.090%), '12333': 923(0.090%), '12334': 984(0.100%), '12341': 1022(0.100%), '12342': 948(0.090%), '12343': 960(0.100%), '12344': 939(0.090%)
'12411': 1011(0.100%), '12412': 996(0.100%), '12413': 991(0.100%), '12414': 944(0.090%), '12421': 989(0.100%), '12422': 1004(0.100%), '12423': 963(0.100%), '12424': 988(0.100%)
'12431': 899(0.090%), '12432': 956(0.100%), '12433': 986(0.100%), '12434': 984(0.100%), '12441': 976(0.100%), '12442': 998(0.100%), '12443': 956(0.100%), '12444': 976(0.100%)
'13111': 995(0.100%), '13112': 987(0.100%), '13113': 1022(0.100%), '13114': 1014(0.100%), '13121': 946(0.090%), '13122': 962(0.100%), '13123': 1024(0.100%), '13124': 961(0.100%)
'13131': 978(0.100%), '13132': 969(0.100%), '13133': 965(0.100%), '13134': 947(0.090%), '13141': 941(0.090%), '13142': 963(0.100%), '13143': 958(0.100%), '13144': 946(0.090%)
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'42331': 962(0.100%), '42332': 986(0.100%), '42333': 975(0.100%), '42334': 983(0.100%), '42341': 1004(0.100%), '42342': 1001(0.100%), '42343': 966(0.100%), '42344': 969(0.100%)
'42411': 1004(0.100%), '42412': 1006(0.100%), '42413': 1035(0.100%), '42414': 928(0.090%), '42421': 1036(0.100%), '42422': 972(0.100%), '42423': 982(0.100%), '42424': 941(0.090%)
'42431': 988(0.100%), '42432': 979(0.100%), '42433': 1002(0.100%), '42434': 924(0.090%), '42441': 953(0.100%), '42442': 990(0.100%), '42443': 1000(0.100%), '42444': 967(0.100%)
'43111': 989(0.100%), '43112': 982(0.100%), '43113': 1004(0.100%), '43114': 939(0.090%), '43121': 970(0.100%), '43122': 1025(0.100%), '43123': 960(0.100%), '43124': 950(0.100%)
'43131': 967(0.100%), '43132': 981(0.100%), '43133': 965(0.100%), '43134': 1007(0.100%), '43141': 899(0.090%), '43142': 978(0.100%), '43143': 937(0.090%), '43144': 952(0.100%)
'43211': 976(0.100%), '43212': 966(0.100%), '43213': 952(0.100%), '43214': 955(0.100%), '43221': 1026(0.100%), '43222': 971(0.100%), '43223': 984(0.100%), '43224': 938(0.090%)
'43231': 940(0.090%), '43232': 971(0.100%), '43233': 1006(0.100%), '43234': 961(0.100%), '43241': 991(0.100%), '43242': 1037(0.100%), '43243': 1048(0.100%), '43244': 933(0.090%)
'43311': 982(0.100%), '43312': 995(0.100%), '43313': 967(0.100%), '43314': 1001(0.100%), '43321': 987(0.100%), '43322': 973(0.100%), '43323': 964(0.100%), '43324': 930(0.090%)
'43331': 886(0.090%), '43332': 1007(0.100%), '43333': 968(0.100%), '43334': 929(0.090%), '43341': 993(0.100%), '43342': 1000(0.100%), '43343': 953(0.100%), '43344': 983(0.100%)
'43411': 1003(0.100%), '43412': 948(0.090%), '43413': 960(0.100%), '43414': 995(0.100%), '43421': 982(0.100%), '43422': 965(0.100%), '43423': 953(0.100%), '43424': 967(0.100%)
'43431': 941(0.090%), '43432': 993(0.100%), '43433': 990(0.100%), '43434': 1028(0.100%), '43441': 986(0.100%), '43442': 930(0.090%), '43443': 1044(0.100%), '43444': 1009(0.100%)
'44111': 1002(0.100%), '44112': 983(0.100%), '44113': 1040(0.100%), '44114': 998(0.100%), '44121': 964(0.100%), '44122': 979(0.100%), '44123': 930(0.090%), '44124': 962(0.100%)
'44131': 1009(0.100%), '44132': 973(0.100%), '44133': 989(0.100%), '44134': 957(0.100%), '44141': 986(0.100%), '44142': 987(0.100%), '44143': 925(0.090%), '44144': 941(0.090%)
'44211': 952(0.100%), '44212': 921(0.090%), '44213': 1031(0.100%), '44214': 972(0.100%), '44221': 963(0.100%), '44222': 985(0.100%), '44223': 969(0.100%), '44224': 997(0.100%)
'44231': 961(0.100%), '44232': 942(0.090%), '44233': 980(0.100%), '44234': 1036(0.100%), '44241': 1034(0.100%), '44242': 970(0.100%), '44243': 992(0.100%), '44244': 938(0.090%)
'44311': 933(0.090%), '44312': 1023(0.100%), '44313': 1030(0.100%), '44314': 1013(0.100%), '44321': 961(0.100%), '44322': 955(0.100%), '44323': 994(0.100%), '44324': 1001(0.100%)
'44331': 978(0.100%), '44332': 967(0.100%), '44333': 951(0.100%), '44334': 963(0.100%), '44341': 998(0.100%), '44342': 990(0.100%), '44343': 962(0.100%), '44344': 973(0.100%)
'44411': 969(0.100%), '44412': 919(0.090%), '44413': 939(0.090%), '44414': 974(0.100%), '44421': 998(0.100%), '44422': 948(0.090%), '44423': 944(0.090%), '44424': 949(0.090%)
'44431': 997(0.100%), '44432': 970(0.100%), '44433': 1022(0.100%), '44434': 975(0.100%), '44441': 994(0.100%), '44442': 989(0.100%), '44443': 982(0.100%), '44444': 993(0.100%)
Question 1 asked 1000000
Answer 1 correct: 249931 - 24.990%
Answer 2 correct: 250264 - 25.030%
Answer 3 correct: 249748 - 24.970%
Answer 4 correct: 250057 - 25.010%
Question 2 asked 1000000
Answer 1 correct: 250300 - 25.030%
Answer 2 correct: 250013 - 25.000%
Answer 3 correct: 250055 - 25.010%
Answer 4 correct: 249632 - 24.960%
Question 3 asked 1000000
Answer 1 correct: 249916 - 24.990%
Answer 2 correct: 250063 - 25.010%
Answer 3 correct: 249698 - 24.970%
Answer 4 correct: 250323 - 25.030%
Question 4 asked 1000000
Answer 1 correct: 250259 - 25.030%
Answer 2 correct: 250025 - 25.000%
Answer 3 correct: 249627 - 24.960%
Answer 4 correct: 250089 - 25.010%
Question 5 asked 1000000
Answer 1 correct: 249525 - 24.950%
Answer 2 correct: 250385 - 25.040%
Answer 3 correct: 249702 - 24.970%
Answer 4 correct: 250388 - 25.040%
_________________
WAS IST LOS IN EUROPA? https://www.youtube.com/watch?v=a9mduhSSC5w |
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Myon
Anmeldungsdatum: 04.12.2013
Beiträge: 5972
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| Myon Verfasst am: 11. Mai 2024 20:59 Titel: |
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Zu den "Dreierkombinationen" - also Folge richtiger Antworten (Schema ABC, ABD, ABE, ..): Ja, es gibt 10 mögliche Folgen von genau 3 richtigen Antworten aus 5 Fragen. Zu jeder dieser Folgen gibt es 3^2 "Elementarereignisse", denn für die 2 falschen Antworten gibt es jeweils drei Möglichkeiten. Somit gibt es insgesamt 10*3^2 günstige Ereignisse aus total 4^5. Die Wahrscheinlichkeit, zufällig genau 3 Fragen richtig zu beantworten ist somit
=\frac{\text{Anzahl günstige Ereignisse}}{\text{Gesamtzahl der Ereignisse}}=\frac{10\cdot 3^2}{4^5}=8.79%)
Das gleiche Ergebenis erhält man mit folgender Überlegung: Die Wahrscheinlichkeit für eine bestimmte Antwortkombination mit genau 3 richtigen Antworten (z.B. "A richtig, B falsch, C richtig, D richtig, E falsch") beträgt
^3\cdot (\frac{3}{4})^2)
Nun multipliziert man das mit der Anzahl Möglichkeiten, 3 aus 5 Fragen auszuwählen (Kombination mit Wiederholung, wieder die 10 Möglichkeiten):
=\begin{pmatrix}5\\3\end{pmatrix}(\frac{1}{4})^3\cdot(\frac{3}{4})^2=8.79%)
In der ursprünglichen Aufgabenstellung ging es um die Wahrscheinlichkeit von mindestens 3 richtigen Antworten. Es wären also noch die Wahrscheinlichkeiten für genau 4 und genau 5 richtige Antworten zu addieren.
Übrigens: Schön, dass Du auch wieder einmal hier im Forum unterwegs bist! |
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VeryApe
Anmeldungsdatum: 10.02.2008
Beiträge: 3273
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| VeryApe Verfasst am: 12. Mai 2024 20:10 Titel: |
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| Zitat: |
Ja, es gibt 10 mögliche Folgen von genau 3 richtigen Antworten aus 5 Fragen. Zu jeder dieser Folgen gibt es 3^2 "Elementarereignisse", denn für die 2 falschen Antworten gibt es jeweils drei Möglichkeiten. Somit gibt es insgesamt 10*3^2 günstige Ereignisse aus total 4^5
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das ist sogar noch besser, weil einleuchtender und man hat keine Verzahnung der Möglichkeiten ineinander.
Wenn ich jetzt nur 4 Fragen beantworte, seien es ABCD und die E Frage beantworte ich zwar auch aber zähle ich nicht, wenn ich treffe.
Dann sind die möglichen dreier Kombinationen ABC, ABD, ACD, BCD dazu 4*3*2/(2*3)
dann habe ich 3^2 möglichkeiten, wo genau jede dreier Kombination trifft und die anderen zwei nicht treffen.
 Wahrscheinlichkeit, das genau eine dreier Kombination trifft, und das aber 4 mal für alle vier Möglichkeiten
nehme ich jetzt noch die Möglichkeiten wo ABCD trifft hinzu, sind es 4 Möglichkeiten, und da ist der Treffer ABCDE mit dabei.
dann fehlen nur noch 3mal ABC_E(wo D nicht trifft), 3mal ABD_E (wo C nicht trifft), 3 mal ACD_E (wo B nicht trifft), 3mal BCD E(wo A nicht trifft).
und ich bin bei
}{4^5}=0,05078125) siehe oben für 4 getätige Antworten und drei müssen treffen.
Das ist halt einleuchtender als dieser Verteilsatz.
MAn sieht auch schön die Verzahnung.
Denn betrachtet man die 9 Möglichkeiten wo jede dreier Kombination genau trifft und man addiert die Möglichkeiten für ABCD..
so bringt das in jeder Gruppe
ABC ABD ADC BCD 4 Möglichkeiten hinzu, wodurch jede Gruppe nun 13 Möglichkeiten hat. Diese Möglichkeiten sind aber ineinader verzahnt, man darf sie nur einmal zählen nicht 4mal. weils die selben Möglichkeiten sind nur unterschiedlich in jeder Gruppe betrachtet.
Ich war vorher die ganze Zeit auf Unix stackexchange und stackoverflow bzw Android, dann gings halt mehr ums programmieren und Linux.
_________________
WAS IST LOS IN EUROPA? https://www.youtube.com/watch?v=a9mduhSSC5w |
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