• The "Kebab Palace Casino" in Madsdale has dice with 20 sides.
• The "house" rolls two 20-sided dice and the "player" rolls one 20-sided die.
• If the player rolls a number on his die between the two numbers the house rolled, then the player wins.
• Otherwise, the house wins (including ties).
Determine the player's probability of winning.
All the equally likely possible conbinations of the two house dice are shown with the number of possible winning rolls of the player's die:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
3 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
5 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
6 4 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13
7 5 4 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12
8 6 5 4 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11
9 7 6 5 4 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10
10 8 7 6 5 4 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9
11 9 8 7 6 5 4 3 2 1 0 0 0 1 2 3 4 5 6 7 8
12 10 9 8 7 6 5 4 3 2 1 0 0 0 1 2 3 4 5 6 7
13 11 10 9 8 7 6 5 4 3 2 1 0 0 0 1 2 3 4 5 6
14 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0 1 2 3 4 5
15 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0 1 2 3 4
16 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0 1 2 3
17 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0 1 2
18 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0 1
19 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0
20 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0
These 400 numbers add up to 2280, making the average 5.7. That is, on average, only 5.7 of the 20 possible numbers on the player's die will be a win for the player. That gives him a probability of 5.7/20 = 28.5%.
If the house didn't win on a tie but the player did, this would be
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
3 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
4 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
5 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
7 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14
8 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13
9 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12
10 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11
11 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10
12 12 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9
13 13 12 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8
14 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7
15 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6
16 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5
17 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 3 4
18 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 3
19 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2
20 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
tot =
3060
giving an average of 7.65 for a probability of a win being 7.65/400 = 38.25 %.
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Posted by Charlie
on 2023-04-04 12:58:30 |
Counting the ways another way
