document.write( "Question 1197440: There is a gambling game where equal-weight balls with numbers ranging from 1 to 15 are placed inside a tambiolo and mixed. Without replacement, three balls are chosen at random and are then chosen one by one.\r
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\n" ); document.write( "\n" ); document.write( "a. Mr. Lewis always bets that none of the 3 balls will include consecutive digits in the outcome. How likely is it that something will occur?\r
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\n" ); document.write( "\n" ); document.write( "b. As long as he wins 9 bets, Mr. Lewis will have enough money to continue playing the game. What is the chance that he will only place a total of 12 bets? \n" ); document.write( "
Algebra.Com's Answer #830810 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I can only help you with the (a) part.\r\n" );
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document.write( "(a) We calculate the number of ways to get a successful choice of 3 balls.\r\n" );
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document.write( "There are 15C3 = [(15)(14)(13)]/[(3)(2)(1)] = 455 ways to choose any 3 balls.\r\n" );
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document.write( "From those 455, we will subtract the number of unsuccessful choices,\r\n" );
document.write( "i.e., those containing a pair of consecutive choices.\r\n" );
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document.write( "There are 14 pairs of consecutively numbered balls: {1,2},{2,3},...,{14,15}\r\n" );
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document.write( "For each of those pairs of balls we can select a third ball to go with it\r\n" );
document.write( "in 13 ways.  That's (14)(13) = 182 ways.\r\n" );
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document.write( "However, among that 182, we have counted each case of the following 13 choices\r\n" );
document.write( "twice each:\r\n" );
document.write( "{1,2,3},{2,3,4},...{13,14,15}\r\n" );
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document.write( "To show that, take for example the choice {6,7,8}.  It was counted once when\r\n" );
document.write( "we put 8 with the pair {6,7} and again when we put 6 with the pair {7,8}.\r\n" );
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document.write( "So we subtract to get 182-13 = 169 unsuccessful choices. \r\n" );
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document.write( "We subtract these 169 ways from the 455 to get 455-169=286 possible successful\r\n" );
document.write( "choices.\r\n" );
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document.write( "So the probability that Mr. Lewis makes a successful choice is 286/455 = 22/35\r\n" );
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document.write( "(b) I suspect I do not not understand the game well enough to do the (b) part.\r\n" );
document.write( "As I see it, there is not enough information.  We are not told how much money he\r\n" );
document.write( "wins or how much it costs to place a bet.  As I see it, the problem will be\r\n" );
document.write( "quite different if he walks into the casino with millions of dollars in his\r\n" );
document.write( "pocket than if he walks in with only enough money to play once, and wins enough\r\n" );
document.write( "each time to continue playing 12 times.  So it looks to me as if we would also\r\n" );
document.write( "need to know how much he wins per successful bet.  As I said, I suspect I do not\r\n" );
document.write( "understand the game well enough to answer the (b) part.  Perhaps another tutor\r\n" );
document.write( "who understands the game will jump in here.  Otherwise post again, explaining\r\n" );
document.write( "how the game operates a little better.  \r\n" );
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document.write( "Edwin

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