document.write( "Question 1199787: You roll two six-sided fair dice.\r
\n" ); document.write( "\n" ); document.write( "a. Let A be the event that either a 4 or 5 is rolled first followed by an even number.
\n" ); document.write( " P(A) =
\n" ); document.write( " Round your answer to four decimal places.\r
\n" ); document.write( "\n" ); document.write( "b. Let B be the event that the sum of the two dice is at most 5.
\n" ); document.write( " P(B) =
\n" ); document.write( " Round your answer to four decimal places. \n" ); document.write( "

Algebra.Com's Answer #833756 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "Part (a)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Define these two events:
\n" ); document.write( "C = event of rolling a 4 or 5 on a single die
\n" ); document.write( "D = event of rolling an even number on a single die
\n" ); document.write( "Each roll of the die is independent.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "P(C) = 2/6 = 1/3
\n" ); document.write( "P(D) = 3/6 = 1/2\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "P(A) = P(C and D)
\n" ); document.write( "P(C and D) = P(C)*P(D) ... independent events
\n" ); document.write( "P(C and D) = (1/3)*(1/2)
\n" ); document.write( "P(C and D) = 1/6
\n" ); document.write( "P(A) = 1/6 = 0.1667
\n" ); document.write( "The 6's go on forever in the decimal value, but we round to four decimal places.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer: 0.1667\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "====================================================================\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Part (b)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "B = event of the sum is at most 5
\n" ); document.write( "B = event the sum is 5 or smaller
\n" ); document.write( "B = event of the sum being in the set {2,3,4,5}\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Sum of Dice Chart
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

\n" ); document.write( "Example calculation: rolling 1 on the blue die, and rolling a 6 on the red die gives a sum of 1+6 = 7 (upper right corner)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The goal is to get a sum of 5 or smaller.
\n" ); document.write( "Focus on the following subset of sums from that chart
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\r\n" );
document.write( "1+1 = 2\r\n" );
document.write( "1+2 = 3\r\n" );
document.write( "1+3 = 4\r\n" );
document.write( "1+4 = 5\r\n" );
document.write( "............\r\n" );
document.write( "2+1 = 3\r\n" );
document.write( "2+2 = 4\r\n" );
document.write( "2+3 = 5\r\n" );
document.write( "............\r\n" );
document.write( "3+1 = 4\r\n" );
document.write( "3+2 = 5\r\n" );
document.write( "............\r\n" );
document.write( "4+1 = 5\r\n" );
document.write( "

\n" ); document.write( "Take note how I'm breaking up the calculations.
\n" ); document.write( "The first group involves \"1\" as the first value, the second group involves \"2\" as the first value, and so on.
\n" ); document.write( "This allows for a predictable pattern and structure.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "There are 4 items in the first group, 3 in the second, 2 in the third, and 1 in the fourth.
\n" ); document.write( "4+3+2+1 = 10 sums in which belong to the set {2,3,4,5}\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "There are 10 instances of achieving event B out of 6*6 = 36 ways to roll two dice.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "10/36 = (2*5)/(2*18) = 5/18 is the probability that event B occurs.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Use a calculator or long division to find that:
\n" ); document.write( "5/18 = 0.2778
\n" ); document.write( "The 7's go on forever, but we round to four decimal places.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer: 0.2778
\n" ); document.write( " \n" ); document.write( "

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