document.write( "Question 1196729: Construct a probability distribution for the sum shown on the faces when two dice, each with 7 faces, are rolled.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #829709 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Here is what the addition table looks like when we add values from two 7-sided dice\n" ); document.write( "\n" ); document.write( "

+	1	2	3	4	5	6	7
1	2	3	4	5	6	7	8
2	3	4	5	6	7	8	9
3	4	5	6	7	8	9	10
4	5	6	7	8	9	10	11
5	6	7	8	9	10	11	12
6	7	8	9	10	11	12	13
7	8	9	10	11	12	13	14

For example, if we roll a 1 on a blue die and a 7 on a red die, then we get 1+7 = 8 as shown in the upper right corner of that table.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The possible sums are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "X = sum of the two 7-sided dice
\n" ); document.write( "P(X) = probability of that sum showing up\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The X values range from 2 to 14 inclusive.
\n" ); document.write( "The sum \"2\" shows up exactly once out of 7*7 = 49 possible outcomes. Therefore P(X) = 1/49 when X = 2
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

X	P(X)
2	1/49
3
4
5
6
7
8
9
10
11
12
13
14

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Then we have the sum \"3\" show up twice out of 49 possible outcomes. We write 2/49 next to 3 like so
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

X	P(X)
2	1/49
3	2/49
4
5
6
7
8
9
10
11
12
13
14

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Keep this process going until you have this completed table.\n" ); document.write( "\n" ); document.write( "

X	P(X)
2	1/49
3	2/49
4	3/49
5	4/49
6	5/49
7	6/49
8	7/49
9	6/49
10	5/49
11	4/49
12	3/49
13	2/49
14	1/49

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You could write the table out like this if you prefer\n" ); document.write( "\n" ); document.write( "

X	2	3	4	5	6	7	8	9	10	11	12	13	14
P(X)	1/49	2/49	3/49	4/49	5/49	6/49	7/49	6/49	5/49	4/49	3/49	2/49	1/49

A pattern to notice is the numerators start to increase {1,2,3,4,5,6,7} when going from X = 2 to X = 8; afterward we have a decreasing set of numerators {6,5,4,3,2,1}
\n" ); document.write( "Optionally you could reduce 7/49 to get 1/7, but I find it's better to have all the denominators be the same.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Side note: all of the P(X) values are between 0 and 1. Also, the P(X) values sum to 1.
\n" ); document.write( " \n" ); document.write( "

\n" );