document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=982389>Question 982389</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two dice are each numbered from 1 to 6, but are biased so that each is twice as likely to land on any of the even numbers as on any of the odd numbers. The two dice are rolled and the numbers are multiplied together. What is the probability that the resulting product is odd?<BR>\n" );
document.write( "A. 1/9<BR>\n" );
document.write( "B. 2/9<BR>\n" );
document.write( "C. 1/3<BR>\n" );
document.write( "D. 4/9<BR>\n" );
document.write( "E. 2/3<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 #603268 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=603268\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Focus on just one die. Let x = the probability of rolling an odd number\r<BR>\n" );
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document.write( "In notation form, x = P(odd)\r<BR>\n" );
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document.write( "So P(even) = 2*P(odd) = 2x because rolling an even number is twice as likely compared to rolling an odd number.\r<BR>\n" );
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document.write( "You can only roll an even number or an odd number. Nothing else. So the probabilities must add to 1\r<BR>\n" );
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document.write( "P(even) + P(odd) = 1<BR>\n" );
document.write( "2x + x = 1<BR>\n" );
document.write( "3x = 1<BR>\n" );
document.write( "x = 1/3\r<BR>\n" );
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document.write( "So we know that rolling an odd number has the probability of 1/3, while the probability of rolling an even number is 2*(1/3) = 2/3.\r<BR>\n" );
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document.write( "Now we turn to these properties of integers\r<BR>\n" );
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document.write( "(even)*(even) = even<BR>\n" );
document.write( "(odd)*(even) = even<BR>\n" );
document.write( "(even)*(odd) = even<BR>\n" );
document.write( "(odd)*(odd) = odd\r<BR>\n" );
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document.write( "There's only one case where you get an odd product: when the two individual dies are odd themselves. \r<BR>\n" );
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document.write( "When they ask \"What is the probability that the resulting product is odd?\" they are equivalently asking \"What is the probability that BOTH dies are odd?\"\r<BR>\n" );
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document.write( "Going back to notation form\r<BR>\n" );
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document.write( "P(odd product) = P(both dies are odd)<BR>\n" );
document.write( "P(odd product) = P(die A is odd AND die B is odd)<BR>\n" );
document.write( "P(odd product) = P(die A is odd)*P(die B is odd) ... see Note below<BR>\n" );
document.write( "P(odd product) = (1/3)*(1/3)<BR>\n" );
document.write( "P(odd product) = (1*1)/(3*3)<BR>\n" );
document.write( "P(odd product) = <font color=\"red\">1/9</font>\r<BR>\n" );
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document.write( "The probability of rolling the 2 biased dice, and have their product be an odd number, is <font color=\"red\">1/9</font>. Of course, this answer would be different if the dice weren't biased.\r<BR>\n" );
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document.write( "Note: The two dice may be crooked, but they are still independent. What this means is that one die does not affect the other. If events x and y are independent, we can say P(x AND y) = P(x) * P(y) which is applied above. </SPAN>\n" );
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