document.write( "Question 1195519: In the 6/49 lotto game, suppose that the minimum prize is 20 pesos, which you can win by guessing 3 out of 6 numbers correctly. The prize goes up as you guess more numbers correctly. What is the probability of not winning prizes of any amount? \n" ); document.write( "

Algebra.Com's Answer #828033 by ikleyn(52797) $\"\"$ $\"About$

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\n" ); document.write( "In the 6/49 lotto game, suppose that the minimum prize is 20 pesos,
\n" ); document.write( "which you can win by guessing 3 out of 6 numbers correctly.
\n" ); document.write( "The prize goes up as you guess more numbers correctly.
\n" ); document.write( "What is the probability of not winning prizes of any amount?
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\n" ); document.write( "\n" ); document.write( "Surely, such problem should start with clear formal description of this
\n" ); document.write( "lotto game - - - but this one does not.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, I will give my own description, as I understand the problem.\r
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document.write( "      Six numbers are drawn from a set of 49 different numbers 1,2,3,...,49.\r\n" );
document.write( "      If a ticket matches at least three numbers, the ticket wins.\r\n" );
document.write( "      The prize goes up as you guess more numbers correctly. \r\n" );
document.write( "      What is the probability of not winning prizes of any amount?\r\n" );
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Solution

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document.write( "First, let's calculate the number of all issued/(possibly existing) tickets.\r\n" );
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document.write( "The total number of all issued/(possibly existing) tickets is the number \r\n" );
document.write( "of combinations of 49 items taken 6 at a time. It is the number of combinations\r\n" );
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document.write( "     =  = 13983816.\r\n" );
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document.write( "Favorable (winning) tickets are those that carry at least 3 of 6 different drawn numbers.\r\n" );
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document.write( "In accordance to it, UNWINNING tickets are those that \r\n" );
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document.write( "    - (a) carry zero of 6 drawn numbers;\r\n" );
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document.write( "    - (b) carry only one of 6 drawn numbers;\r\n" );
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document.write( "    - (c) carry only two of 6 drawn numbers.\r\n" );
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document.write( "So, we divide 49 numbers in two categories: 6 selected (drawn) numbers and the rest 49-6 = 43 numbers.\r\n" );
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document.write( "In category (a), we have    tickets;\r\n" );
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document.write( "In category (b), we have    tickets;\r\n" );
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document.write( "In category (c), we have    tickets.\r\n" );
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document.write( "    +---------------------------------------------------------------------+\r\n" );
document.write( "    |    Notice that the order of numbers in tickets does not matter -    |\r\n" );
document.write( "    |               - therefore we use combinations.                      |\r\n" );
document.write( "    +---------------------------------------------------------------------+\r\n" );
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document.write( "These three categories are DISJOINT - - - therefore, the total of UNWINNING tickets is the sum\r\n" );
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document.write( "     +  +  = 6096454 + 6*962598 + 15*123410 = 13723192.\r\n" );
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document.write( "Now the last step to find the probability of getting unwinning ticket is this fraction\r\n" );
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document.write( "    P =  =  = 0.981362455.\r\n" );
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document.write( "ANSWER.  The probability of getting unwinning ticket is 0.98126 (rounded).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "It is a good problem of an advanced level.\r
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