SOLUTION: please help me solve the following question “A fair die is thrown at random three independent times. Let Y be the maximum of these three outcomes, each outcome being the number

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Question 998529: please help me solve the following question
“A fair die is thrown at random three independent times. Let Y be the maximum of these three outcomes, each outcome being the number shown on the face uppermost on the die. Find the probability density function of Y”.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
I used EXCEL to generate all 216+%283%2A3%2A3%29 possible outcomes and sums.
1 1 1 3
1 1 2 4
1 1 3 5
1 1 4 6
1 1 5 7
1 1 6 8
1 2 1 4
1 2 2 5
1 2 3 6
1 2 4 7
1 2 5 8
1 2 6 9
1 3 1 5
1 3 2 6
1 3 3 7
1 3 4 8
1 3 5 9
1 3 6 10
1 4 1 6
1 4 2 7
1 4 3 8
1 4 4 9
1 4 5 10
1 4 6 11
1 5 1 7
1 5 2 8
1 5 3 9
1 5 4 10
1 5 5 11
1 5 6 12
1 6 1 8
1 6 2 9
1 6 3 10
1 6 4 11
1 6 5 12
1 6 6 13
2 1 1 4
2 1 2 5
2 1 3 6
2 1 4 7
2 1 5 8
2 1 6 9
2 2 1 5
2 2 2 6
2 2 3 7
2 2 4 8
2 2 5 9
2 2 6 10
2 3 1 6
2 3 2 7
2 3 3 8
2 3 4 9
2 3 5 10
2 3 6 11
2 4 1 7
2 4 2 8
2 4 3 9
2 4 4 10
2 4 5 11
2 4 6 12
2 5 1 8
2 5 2 9
2 5 3 10
2 5 4 11
2 5 5 12
2 5 6 13
2 6 1 9
2 6 2 10
2 6 3 11
2 6 4 12
2 6 5 13
2 6 6 14
3 1 1 5
3 1 2 6
3 1 3 7
3 1 4 8
3 1 5 9
3 1 6 10
3 2 1 6
3 2 2 7
3 2 3 8
3 2 4 9
3 2 5 10
3 2 6 11
3 3 1 7
3 3 2 8
3 3 3 9
3 3 4 10
3 3 5 11
3 3 6 12
3 4 1 8
3 4 2 9
3 4 3 10
3 4 4 11
3 4 5 12
3 4 6 13
3 5 1 9
3 5 2 10
3 5 3 11
3 5 4 12
3 5 5 13
3 5 6 14
3 6 1 10
3 6 2 11
3 6 3 12
3 6 4 13
3 6 5 14
3 6 6 15
4 1 1 6
4 1 2 7
4 1 3 8
4 1 4 9
4 1 5 10
4 1 6 11
4 2 1 7
4 2 2 8
4 2 3 9
4 2 4 10
4 2 5 11
4 2 6 12
4 3 1 8
4 3 2 9
4 3 3 10
4 3 4 11
4 3 5 12
4 3 6 13
4 4 1 9
4 4 2 10
4 4 3 11
4 4 4 12
4 4 5 13
4 4 6 14
4 5 1 10
4 5 2 11
4 5 3 12
4 5 4 13
4 5 5 14
4 5 6 15
4 6 1 11
4 6 2 12
4 6 3 13
4 6 4 14
4 6 5 15
4 6 6 16
5 1 1 7
5 1 2 8
5 1 3 9
5 1 4 10
5 1 5 11
5 1 6 12
5 2 1 8
5 2 2 9
5 2 3 10
5 2 4 11
5 2 5 12
5 2 6 13
5 3 1 9
5 3 2 10
5 3 3 11
5 3 4 12
5 3 5 13
5 3 6 14
5 4 1 10
5 4 2 11
5 4 3 12
5 4 4 13
5 4 5 14
5 4 6 15
5 5 1 11
5 5 2 12
5 5 3 13
5 5 4 14
5 5 5 15
5 5 6 16
5 6 1 12
5 6 2 13
5 6 3 14
5 6 4 15
5 6 5 16
5 6 6 17
6 1 1 8
6 1 2 9
6 1 3 10
6 1 4 11
6 1 5 12
6 1 6 13
6 2 1 9
6 2 2 10
6 2 3 11
6 2 4 12
6 2 5 13
6 2 6 14
6 3 1 10
6 3 2 11
6 3 3 12
6 3 4 13
6 3 5 14
6 3 6 15
6 4 1 11
6 4 2 12
6 4 3 13
6 4 4 14
6 4 5 15
6 4 6 16
6 5 1 12
6 5 2 13
6 5 3 14
6 5 4 15
6 5 5 16
6 5 6 17
6 6 1 13
6 6 2 14
6 6 3 15
6 6 4 16
6 6 5 17
6 6 6 18
So then adding up the possible outcomes from 3 to 18.
3 1
4 3
5 6
6 10
7 15
8 21
9 25
10 27
11 27
12 25
13 21
14 15
15 10
16 6
17 3
18 1
and then dividing the outcomes by the total number of outcomes,
3 0.0046
4 0.0139
5 0.0278
6 0.0463
7 0.0694
8 0.0972
9 0.1157
10 0.1250
11 0.1250
12 0.1157
13 0.0972
14 0.0694
15 0.0463
16 0.0278
17 0.0139
18 0.0046