document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197565>Question 1197565</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pair of 6 sided dice are tossed. What is the probability that at least one of the dice has a value greater than or equal to 5? </SPAN>\n" );
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document.write( "There are 6 sides on each dice, giving 6*6 = 36 outcomes.\r<BR>\n" );
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document.write( "Let's call the dice A and B<BR>\n" );
document.write( "Let's have die A roll to 4 or smaller. Same goes for die B.<BR>\n" );
document.write( "There are 4*4 = 16 outcomes where both dice are 4 or smaller.\r<BR>\n" );
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document.write( "That must mean 36-16 = 20 outcomes are when either <BR>\n" );
document.write( "A = 5 or 6<BR>\n" );
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document.write( "B = 5 or 6<BR>\n" );
document.write( "or both scenarios happen\r<BR>\n" );
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document.write( "We can confirm this with a chart<BR>\n" );
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document.write( "<tr><TD ></TD><TD >1</TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD><TD >6</TD></tr><tr><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD><TD >X</TD></tr><tr><TD >2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD><TD >X</TD></tr><tr><TD >3</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD><TD >X</TD></tr><tr><TD >4</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD><TD >X</TD></tr><tr><TD >5</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD></tr><tr><TD >6</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD></tr>\n" );
document.write( "</TABLE>The 'X's represent the situation when either dice is 5 or larger (i.e. 5 or 6) \r<BR>\n" );
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document.write( "There are 6 X's in row 5 and 6 X's in row 6<BR>\n" );
document.write( "That's 6+6 = 12 X's total so far.<BR>\n" );
document.write( "We have 6 X's in column 5 and 6 X's in column 6<BR>\n" );
document.write( "That's another 6+6 = 12 X's<BR>\n" );
document.write( "We're up to 12+12 = 24 \r<BR>\n" );
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document.write( "But in the bottom right hand corner, we're double counting those four X's since they are in the overlapped region of the four columns mentioned.<BR>\n" );
document.write( "We need to subtract off 4 to correct for this erroneous double counting.<BR>\n" );
document.write( "24-4 = 20\r<BR>\n" );
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document.write( "Or note that the blank portion of the table is 4 rows tall and 4 columns across.<BR>\n" );
document.write( "This represents the 4*4 = 16 cases where both dice are 4 or smaller, as mentioned earlier.<BR>\n" );
document.write( "This then gives us the 36-16 = 20 copies of X in total.\r<BR>\n" );
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document.write( "----------------------------------------------------\r<BR>\n" );
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document.write( "The summary so far is that there are 20 ways to have at least one of the dice be 5 or 6.\r<BR>\n" );
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document.write( "This is out of 36 ways to roll two dice<BR>\n" );
document.write( "20/36 = (4*5)/(4*9) = 5/9\r<BR>\n" );
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document.write( "Answer: <font color=red>5/9</font><BR>\n" );
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