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%%% file:      s3.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2015/11/02                                                    %%%
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\begin{document}

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\title{3. domáca úloha z predmetu \\ 1-AIN-121 Matematika (1) ZS 2015/16}

\author{Tatiana Jajcayová \and Ján Komara}

%\date{}
\date{\today}

\maketitle

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\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}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\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 skrutky vyberá náhodne, aká je pravdepodobnosť, že si vybral
chybnú skrutku?

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. 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?

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. príklad}

Pomocou binomickej vety nájdite aproximáciu (s presnosťou na $10^{-3}$) hodnoty
$0.99^9$.

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\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

Dokážte tzv. Cauchyho sčítací vzorec (známy tiež pod menom Vandermondova
konvolúcia): pre všetky prirodzené čísla $m$, $n$ a {k} platí
\[
\sum_{i=0}^k \binom{m}{i} \binom{n}{k-i} = \binom{m+n}{k}.
\]

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\end{document}

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