document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=335676>Question 335676</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My Algebra I book has this question.  I know how to solve it, but I want to understand WHY the formula works for finding probability.  Here's the problem:  <BR>\n" );
document.write( "\"When you play a game with number cubes, you can find probabilities by squaring a binomial.  Let A represent rolling 1 or 2 and B represent rolling 2,3,4, or 6.  The probability of A is 1/3 and the probability of B is 2/3.  Find the solution: (1/3A + 2/3B)squared.\"<BR>\n" );
document.write( "Again, I want to understand WHY the formula (1/3A + 2/3B)squared works for identifying probability.  Thank you!!    <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #240716 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=240716\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When you play a game with number cubes, you can find probabilities by squaring a binomial. Let A represent rolling 1 or 2 and B represent rolling 2,3,4, or 6. The probability of A is 1/3 and the probability of B is 2/3. Find the solution: (1/3A + 2/3B)squared.\"<BR>\n" );
document.write( "Again, I want to understand WHY the formula (1/3A + 2/3B)squared works for identifying probability.<BR>\n" );
document.write( "------------------------------<BR>\n" );
document.write( "[(1/3)A + (2/3)B]^2<BR>\n" );
document.write( "----<BR>\n" );
document.write( "= [(1/3)A]^2 + 2[(1/3)A(2/3)B] + [(2/3)B]^2<BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "= (1/9)A^2 + 2(1/3)(2/3)AB + (4/9)B^2<BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "= (1/9)AA + 2(1/3)(2/3)AB + (4.9)BB<BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "This says you can get the event AA one way and the probability is (1/3)(1/3)<BR>\n" );
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document.write( "You can get the event AB or BA in two ways and the probability is (1/3)(2/3)<BR>\n" );
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document.write( "You can get the event BB in 1 way and the probability is (2/3)(2/3)<BR>\n" );
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document.write( "That perfectly describes all the possible results when you roll <BR>\n" );
document.write( "a die twice and it shows you the probability of the different outcomes.<BR>\n" );
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document.write( "You ask why?<BR>\n" );
document.write( "I don't think you can ask that of a mathematical model.<BR>\n" );
document.write( "The only question you can ask is \"How good is the model?\"<BR>\n" );
document.write( "In this case the model is perfect so it is used.<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Note ((1/3)A+(2/3)B)^n would show you all the results of rolling<BR>\n" );
document.write( "your die n times and give you the probability of each result.<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "This model was discovered by Blaise Pascal.  The model he developed<BR>\n" );
document.write( "is called \"Pascal's Triangle\"<BR>\n" );
document.write( "====================================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.\r<BR>\n" );
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