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\partie{Homework \#5}

There are many different tests to check the randomness of a table of digits. In this exercise we will discover two
of them and use them to study whether the digits of $\pi$ can be considered random or not.

\probpart{-- The maximal blocks test}

For this test, the digits are grouped in three-digits blocks. A block of three consecutive digits in the table is said
to be maximal when the middle digit is strictly greater then the two other digits. For example, blocks $142, 486, 053$
are maximal whereas blocks $426$ and $663$ aren't.

\begin{enumerate}
\item How many different three-digits blocks are there ?
\item \begin{enumerate}
      \item If a block is maximal and its tens digit is $6$, what are the possible values for the units digit and the
hundreds digit.
      \item Deduce the number of maximal blocks with the tens digit equal to $6$.
      \end{enumerate}
\item Use the same method to fill out the table below .
	  $$\begin{tabular}{|*{11}{c|}} \hline 
	  Tens digit		& 0	& 1	& 2	& 3	& 4	 & 5	& 6	& 7	 & 8 & 9	 \\
\hline
	  \#\ of maximal blocks	&   &	  &   &   &    &    &36 &    &  &     \\ \hline
	  \end{tabular}$$
    \item Deduce the probability of occurence of a maximal block in a random table of digits.
\end{enumerate}

\probpart{-- The poker test}

One of the best tests to check the randomness of a table of digits is the poker test.\\
Digits are grouped in five-digits blocks, divided into 7 categories, or hands :
\begin{multicols}{2}
\begin{itemize}
	\itemb busts : $abcde$ (exemple : 03251) ;
	\itemb pairs : $aabcd$ (exemple : 80919) ;
	\itemb two pairs : $aabbc$ (exemple : 08608) ;
	\itemb threes : $aaabc$ (exemple : 12151) ;
	\itemb full house : $aaabb$ ;
	\itemb fours : $aaaab$ ;
	\itemb fives : $aaaaa$.
\end{itemize}
\end{multicols}
The order of the digits inside the block is not considered. For example, the blocks $12345$ and $13542$ are the same.
The table below gives the probability of each type of hand.
\begin{center}
\begin{tabular}{|p{3cm}*{7}{|c}|}\hline
Hand	& $abcde$ & $ aabcd$ & $aabbc$ & $aaabc$ & $aaabb$ & $aaaab$ & $aaaaa$ \\ \hline
Probability  & $0.3024$ &  $0.5040$  & $0.1080$  & $0.0720$  &  $0.0090$ & $0.0045$  & $0.0001$ \\ \hline
\end{tabular}
\end{center}

\begin{enumerate}
\item How many different five-digits block are there ?
\item In a no-pair hand, how many possible values are there for $a$, for $b$, for $c$, for $d$ and for $e$ ? Deduce the
number of possible hands of this kind and check the probability given in the table above.
\item Explain the probability given for the fives.
\end{enumerate}





\probpart{-- The digits of  $\pi$}

In this part, we will use the two tests to study the randomness of the digits of $\pi$ appearing after the decimal
point.
Here are the first 1005 digits of $\pi$. \label{decimalespi}.
	{\small\begin{align}
	3,&141\;\;592\;653\;589\;793\;238\;462\;643\;383\;279\; 
	502\;884\;197\;169\;399\;375\;105\;820\;974\;944\; \\
	&592\;307\;816\;406\;286\;208\;998\;628\;034\;825\; 
	342\;117\;067\;982\;148\;086\;513\;282\;306\;647\; \\
	&093\;844\;609\;550\;582\;231\;725\;359\;408\;128\; 
	481\;117\;450\;284\;102\;701\;938\;521\;105\;559\; \\
	&644\;622\;948\;954\;930\;381\;964\;428\;810\;975\; 
	665\;933\;446\;128\;475\;648\;233\;786\;783\;165\; \\
	&271\;201\;909\;145\;648\;566\;923\;460\;348\;610\;
	454\;326\;648\;213\;393\;607\;260\;249\;141\;273\;\\
	&724\;587\;006\;606\;315\;588\;174\;881\;520\;920\;
	962\;829\;254\;091\;715\;364\;367\;892\;590\;360\;\\
	&011\;330\;530\;548\;820\;466\;521\;384\;146\;951\;
	941\;511\;609\;433\;057\;270\;365\;759\;591\;953\;\\
	&092\;186\;117\;381\;932\;611\;793\;105\;118\;548\;
	074\;462\;379\;962\;749\;567\;351\;885\;752\;724\;\\
	&891\;227\;938\;183\;011\;949\;129\;833\;673\;362\;
	440\;656\;643\;086\;021\;394\;946\;395\;224\;737\;\\
	&190\;702\;179\;860\;943\;702\;770\;539\;217\;176\;
	293\;176\;752\;384\;674\;818\;467\;669\;405\;132\;\\%600
	&000\;568\;127\;145\;263\;560\;827\;785\;771\;342\;
	757\;789\;609\;173\;637\;178\;721\;468\;440\;901\;\\%660
	&224\;953\;430\;146\;549\;585\;371\;050\;792\;279\;
	689\;258\;923\;542\;019\;956\;112\;129\;021\;960\;\\%720
	&864\;034\;418\;159\;813\;629\;774\;771\;309\;960\;
	518\;707\;211\;349\;999\;998\;372\;978\;049\;951\;\\%780
	&059\;731\;732\;816\;096\;318\;595\;024\;459\;455\;
	346\;908\;302\;642\;522\;308\;253\;344\;685\;035\;\\%840
	&261\;931\;188\;171\;010\;003\;137\;838\;752\;886\;
	587\;533\;208\;381\;420\;617\;177\;669\;147\;303\;\\%900
	&598\;253\;490\;428\;755\;468\;731\;159\;562\;863\;
	882\;353\;787\;593\;751\;957\;781\;857\;780\;532\;\\%960
	&171\;226\;806\;613\;001\;927\;876\;611\;195\;909\;%990
	216\;420\;198\;938\;095\;\;\;\;\;\;%1005
	\end{align}}
%257 201065485 863278865 936153381 827968230 301952035 301852968 995773622 599413891 249721775 283479131 515574857
%242454150 695950829 533116861 727855889 075098381 754637464 939319255 060400927 701671139 009848824 012858361
\begin{enumerate}
\item Apply the maximal blocks test on this table and give clearly the result.
\item Apply the poker test on this table and give clearly the results.
\item Do you think that this table of digits can be considered as random ?
\item Why did we choose to study $1005$ digits, and not just $1000$ ?
\end{enumerate}



\end{document}
