document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=843064>Question 843064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If you have 3 number cubes, what is the probability that you will roll three of a kind or a sum that is an odd number?<BR>\n" );
document.write( "I have tried to solve this problem but I don't understand how to figure out the probability of rolling 3 of a kind. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #507909 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=507909\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the probability of getting three of a kind of the sum of the three cubes is an odd number is equal to:\r<BR>\n" );
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document.write( "the probability of getting three of a kind<BR>\n" );
document.write( "plus the probability of getting the sum of the cubes equal to an odd number<BR>\n" );
document.write( "minus the probability of getting three of a kind and the sum of the cubes is equal to an odd number.\r<BR>\n" );
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document.write( "the probability of getting three of a kind is equal to:<BR>\n" );
document.write( "1/6 * 1/6 * 1/6 * 6 which is equal to:<BR>\n" );
document.write( "6/216\r<BR>\n" );
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document.write( "out of the 6, 3 of them will have a sum that is odd (numbers 1,3,5) and 3 of them will have a sum that is even (numbers 2,4,6).\r<BR>\n" );
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document.write( "we need to subtract the ones that have an odd sum because they will otherwise be double counted.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "6/216 minus 3/216 which is equal to:<BR>\n" );
document.write( "3/216\r<BR>\n" );
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document.write( "the net probability of getting 3 of a kind whose sum is odd is therefore equal to 3/216. *****\r<BR>\n" );
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document.write( "the probability of the sum of the three cubes having a sum that is odd is broken up into 2 parts.\r<BR>\n" );
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document.write( "the first part is the probability that each of the cubes is odd.  <BR>\n" );
document.write( "this will result in a sum that is odd.\r<BR>\n" );
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document.write( "the probability that each cube is odd is equal to 3/6.<BR>\n" );
document.write( "the probability that all 3 cubes are odd is therefore equal to 3/6 * 3/6 * 3/6 which is equal to 27/216. *****\r<BR>\n" );
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document.write( "note that the probability that each cube is odd includes the cubes that are all the same (cubes with all 1's, cubes with all 3's, cubes with all 5's).   <BR>\n" );
document.write( "if we had not subtracted them from the total, they would have been double counted.<BR>\n" );
document.write( "the subtraction was done above.\r<BR>\n" );
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document.write( "the only other way that the sum of the 3 cubes could be odd is that 2 of the cubes are even and the third cube is odd.\r<BR>\n" );
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document.write( "the probability of a cube being even is 3/6.<BR>\n" );
document.write( "the probability of a cube being odd is 3/6.\r<BR>\n" );
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document.write( "the odd number are 1,3,5<BR>\n" );
document.write( "the even numbers are 2,4,6\r<BR>\n" );
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document.write( "the probability of 2 of the cubes being even and 1 of the cubes being odd is therefore equal to 3 * 3/6 * 3/6 * 3/6 = 81/216 *****\r<BR>\n" );
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document.write( "the 3/6 * 3/6 * 3/6 is only one arrangement.<BR>\n" );
document.write( "there are three arrangements of 2 even cubes and 1 odd cube.\r<BR>\n" );
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document.write( "those arrangements are:\r<BR>\n" );
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document.write( "even even odd<BR>\n" );
document.write( "even odd even<BR>\n" );
document.write( "odd even even\r<BR>\n" );
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document.write( "that's why the 3/6 * 3/6 * 3/6 was also multiplied by 3.<BR>\n" );
document.write( "this takes care of the 3 possible arrangements.\r<BR>\n" );
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document.write( "the total probability is the sum of all 3 of these probabilities.\r<BR>\n" );
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document.write( "that will be 3/216 + 27/216 + 81/216.\r<BR>\n" );
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document.write( "the sum of all of that is equal to 111/216.\r<BR>\n" );
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document.write( "that's your answer is i did this correctly.\r<BR>\n" );
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document.write( "i verified with a much simpler example that the method is good so i'm reasonably confident the answer is good.\r<BR>\n" );
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document.write( "the formula for p(a or b) is:\r<BR>\n" );
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document.write( "p(a or b) = p(a) + p(b) - p(ab)\r<BR>\n" );
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document.write( "p(ab) is the probability of both occurring at the same time.\r<BR>\n" );
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document.write( "in this problem that was the probability that all cubes had the same number and that the sum was odd.\r<BR>\n" );
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document.write( "p(ab) in this problem was 3/216 which i subtracted from 6/216 up front.\r<BR>\n" );
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document.write( "if i had followed the script exactly, the end total would have been:\r<BR>\n" );
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document.write( "6/216 + 27/216 + 81/216 - 3/216.\r<BR>\n" );
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document.write( "the end result would have been the same with a total probability of 114/216 - 3/216 which would have been equal to 111/216.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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