%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%% file:      s6.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2016/12/9                                                     %%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass[12pt,a4paper]{article}

\usepackage[slovak]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{amsmath}
\usepackage{amssymb}

\usepackage{lmodern}
\usepackage{indentfirst}

\renewcommand{\labelenumi}{{\normalfont (\alph{enumi})}}
\renewcommand{\labelenumii}{{\normalfont (\roman{enumii})}}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{6. domáca úloha z predmetu \\ 1-AIN-121 Matematika (1) ZS 2016/17}

\author{Tatiana Jajcayová \and Ján Komara}

%\date{}
\date{\today}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{1. príklad}

Máme šesť krabíc, z toho sú dve okrúhle a štyri štvorcové. Každá okrúhla krabica
obsahuje 2 zelené a 3 modré guličky. Každá štvorcová krabica obsahuje 1 zelenú a
3 modré guličky. Najprv si náhodne zvolíme krabicu, a potom zo zvolenej krabice
náhodne vytiahneme guličku.
\begin{enumerate}
\item 
Aká je pravdepodobnosť, že vytiahnutá gulička je modrá?
\item 
Aká je pravdepodobnosť, že krabica bola okrúhla, ak vieme, že vytiahnutá gulička
je modrá?
\item 
Reprezentujte pravdepodobnosti stromovým diagramom.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{2. príklad}

Stolár používa skrutky vyrobené v dvoch rôznych továrňach, v továrni X a v
továrni Y. 40\% jeho skrutiek pochádza z továrne X, zvyšok je z továrne Y.
Zhruba 4\% skrutiek z továrne X je chybných a zhruba 2\% skrutiek z továrne Y je
chybných. Ak si stolár vyberá skrutky  náhodne, aká je pravdepodobnosť, že si
vybral chybnú skrutku?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{3. príklad}

Basketbalové družstvo má úspešnosť zhruba 1 výhru z každých troch zápasov (t.j.
pravdepodobnosť ich výhry v zápase je $\frac{1}{3}$). V turnaji hrajú presne tri
zápasy.
\begin{enumerate}
\item 
Aká je pravdepodobnosť, že ani raz nevyhrajú?
\item 
Aká je pravdepodobnosť, že vyhrajú tri krát?
\item 
Akú by museli mať úspešnosť, aby ich šanca troch výhier bola viac ako 40\%?.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{4. príklad}

Test na lupus dáva (správne) pozitívne výsledky v 95\% výsledkov, keď sa choroba
naozaj vyskytuje a dáva (nesprávne) pozitívne výsledky v 10\% výsledkov, keď sa
choroba nevyskytuje. Výskyt lupusu v populácii sa odhaduje na 2\%.
\begin{enumerate}
\item 
Ak má náhodne vybratý človek pozitívny test, aká je pravdepodobnosť, že má
lupus?
\item 
Ak má náhodne vybratý človek negatívny test, aká je pravdepodobnosť, že  má
lupus?
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{5. príklad}

Aká je pravdepodobnosť, že na prednáške, na ktorej je 60 študentov a 1
prednášajúca, budú mať dvaja ľudia  narodeniny v ten istý (rovnaký dátum, rok
neuvažujeme) deň? Koľko ľudí by muselo byť v miestnosti, aby bolo rozumné (t.j.
aby bola šanca viac ako 50\%) uzavrieť stávku, že sa tam nachádzajú dvaja ľudia
s narodeninami v ten istý deň?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
