document.write( "Question 1176740: George plays a game using a biased die which is twice as likely to land on
\n" ); document.write( "an even number as on an odd number. The probabilities for the three even
\n" ); document.write( "numbers are all equal and the probabilities for the three odd numbers are
\n" ); document.write( "all equal.
\n" ); document.write( "George throws the die once and calculates his score by the following method.
\n" ); document.write( "• If the number lands on is 3 or less, he multiplies the number by 3 and
\n" ); document.write( "adds 1.
\n" ); document.write( "• If the number lands on is more than 3, he multiplies the number by 2 and
\n" ); document.write( "subtracts 4.\r
\n" ); document.write( "\n" ); document.write( "George throws the die twice.
\n" ); document.write( "(i) Find the probability that the total of the scores on the two throws is
\n" ); document.write( "16.
\n" ); document.write( "(ii) Given that the total of the scores on the two throws is 16, find the
\n" ); document.write( "probability that the score on the first throw was 6. \n" ); document.write( "
Algebra.Com's Answer #804323 by Edwin McCravy(20064)\"\" \"About 
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George plays a game using a biased die which is twice as likely to land on
\n" ); document.write( "an even number as on an odd number. The probabilities for the three even
\n" ); document.write( "numbers are all equal and the probabilities for the three odd numbers are
\n" ); document.write( "all equal.
\n" ); document.write( "
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document.write( "P(1)+P(2)+P(3)+P(4)+P(5)+P(6) = 1\r\n" );
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document.write( "Let the probability of an odd number be x.\r\n" );
document.write( "Then the probability of an even number will be 2x.\r\n" );
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document.write( "x+2x+x+2x+x+2x = 1\r\n" );
document.write( "            9x = 1\r\n" );
document.write( "             x = 1/9\r\n" );
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document.write( "So P(odd) = 1/9 and P(even) = 2/9

\n" ); document.write( "George throws the die once and calculates his score by the following method.
\n" ); document.write( "• If the number lands on is 3 or less, he multiplies the number by 3 and
\n" ); document.write( "adds 1.
\n" ); document.write( "• If the number lands on is more than 3, he multiplies the number by 2 and
\n" ); document.write( "subtracts 4.
\n" ); document.write( "George throws the die twice.
\n" ); document.write( "(i) Find the probability that the total of the scores on the two throws is
\n" ); document.write( "16.
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document.write( "roll  P(roll)  Arithmetic  Score   \r\n" );
document.write( " 1      1/9      1x3+1       4\r\n" );
document.write( " 2      2/9      2x3+1       7\r\n" );
document.write( " 3      1/9      3x3+1      10 \r\n" );
document.write( " 4      2/9      4x2-4       4\r\n" );
document.write( " 5      1/9      5x2-4       6\r\n" );
document.write( " 6      2/9      6x2-4       8 \r\n" );
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document.write( "The only way is for the rolls to be (10 and 6), (6 and 10) or (8 and 8) \r\n" );
document.write( "   \"S1\" = \"score on first roll\"\r\n" );
document.write( "\"S2\" = \"score on second roll\"\r\n" );
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document.write( "P(S1+S2=16) = P[(S1=10 AND S2=6) OR P(S1=6 AND S2=10) OR P(S1=8 and S2=8)] =\r\n" );
document.write( "P(3)∙P(5) + P(5)∙P(3) + P(6)P(6) = (1/9)(1/9) + (1/9)(1/9) + (2/9)(2/9) =\r\n" );
document.write( "1/81+1/81+4/81) = 6/81 which reduces to 2/27.\r\n" );
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(ii) Given that the total of the scores on the two throws is 16, find the
\n" ); document.write( "probability that the score on the first throw was 6.
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document.write( "P(A|B) = P(A and B)/P(B)\r\n" );
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document.write( "P[(S1=10 AND S2=6) OR P(S1=6 AND S2=10) OR P(S1=8 AND S2=8) | P(S1=10)] =\r\n" );
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document.write( "(2/27)/(1/9) = (2/27)(9/1) = 18/27 = 2/3\r\n" );
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document.write( "Edwin
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