document.write( "Question 1177370: . Consider an experiment of choosing randomly two of the letters from the alphabet.
\n" ); document.write( "(a) Find the sample space S1 , of the experiment if it possible to choose the same letter twice.
\n" ); document.write( "(b) Find the sample space S2 , of the experiment if it is not possible to choose the same letter
\n" ); document.write( "twice. \n" ); document.write( "

Algebra.Com's Answer #806375 by ikleyn(52790) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Consider an experiment of choosing randomly two of the letters from the alphabet.
\n" ); document.write( "(a) Find the sample space S1 of the experiment if it possible to choose the same letter twice.
\n" ); document.write( "(b) Find the sample space S2 of the experiment if it is not possible to choose the same letter twice.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            As the \"experiment\" is described in the post, this description is not complete.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            To be complete, the problem must say what is recorded as the experiments' result.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            If you record the letters as the pairs (X,Y), where the order does matter, it is one experiment.\r
\n" ); document.write( "\n" ); document.write( "            If you record the letters as the pairs (X,Y) and the order does not matter, it is another experiment.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            In this another experiment, the pair is (X,Y) considered the same as (Y,X), \r
\n" ); document.write( "\n" ); document.write( "            while in the first experiment these pairs are different.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            As you will see from what follows, the sample spaces are DIFFERENT in these experiments.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            Unfortunately, the problem / (the post) is silent about it, so I will make my own assumptions.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "         * * * Let's consider first the case, when the order does matter. * * * \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "(a)   Find the sample space S1 of the experiment if it possible to choose the same letter twice\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "     Then the sample space S1 is the space of all the pairs  (X,Y), where X and Y are letters \r\n" );
document.write( "     from the alphabet and the order does matter.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     You may think about this sample space S1 as about the (26x26)-matrix of the pairs of the letters.\r\n" );
document.write( "     The size (the number of elements) of the sample space is  26*26 =  = 676 in this case.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "(b) Find the sample space S2 of the experiment if it is not possible to choose the same letter twice\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "     Then the sample space S2 is the space of all the pairs  (X,Y), where X and Y are DIFFERENT letters \r\n" );
document.write( "     from the alphabet and the order does matter.  The pairs of the form (X,X) are not allowed.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     You may think about this sample space S2 as about the (26x26)-matrix of the pairs of the letters WITH EXCLUDED DIAGONAL.\r\n" );
document.write( "     The size (the number of elements of the sample space is  26*26 - 26 =  = 676 - 26 = 650 in this case.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "At this point, the consideration of the first case is FULLY COMPLETED.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " * * * Next, let's consider the case, when the order in pairs does not matter. * * * \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "(a) Find the sample space S1 of the experiment if it possible to choose the same letter twice\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "     Then the sample space S1 is the space of all the pairs  (X,Y), where X and Y are letters \r\n" );
document.write( "     from the alphabet and the order DOES NOT matter:  the pair (X,Y) is the same as (Y,X).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     You may think about this sample space S1 as about the UPPER the DIAGONAL part of the (26x26)-matrix \r\n" );
document.write( "     of the pairs of the letters; the diagonal of the matrix is included to the sample space S1.\r\n" );
document.write( "     The size (the number of elements) of the sample space S1 is   = 13*27 = 351 in this case.\r\n" );
document.write( "

\r\n" );
document.write( "     Then the sample space S2 is the space of all the pairs  (X,Y), where X and Y are DIFFERENT letters \r\n" );
document.write( "     from the alphabet and the order DOES NOT matter.  The pairs of the form (X,X) are not allowed.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     You may think about this sample space S2 as about the UPPER the DIAGONAL part of the (26x26)-matrix of the pairs of the letters \r\n" );
document.write( "     WITH EXCLUDED DIAGONAL of the matrix.\r\n" );
document.write( "     The size (the number of elements) of the sample space S2 is   = 13*27-26 = 351 - 26 = 325 in this case.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "At this point, the solution of the problem is FULLY COMPLETED.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );