October 2013
Problem of the Month
Jackpot Yahtzee
I was playing Jackpot Yahtzee with two of my grandsons recently, and we wondered about the frequency of the different
possible rolls in the game. There are four dice with the following symbols on them:
Dice 1: Orange, Orange, Bell, Bell, Cherry, Dollar
Dice 2: Orange, Orange, Bell, Bell, Cherry, Dollar
Dice 3: Orange, Bell, Dollar, Cherry, Cherry, Cherry
Dice 4: Orange, Bell, Dollar, Cherry, Cherry, Cherry
The best possible roll is to get one of each of the four symbols: Orange, Cherry, Bell, and Dollar.
What is the probability of rolling the four dice and getting one each of the four symbols?
Express your answer as a fraction in reduced form.
Solution to the Problem:
The answer is 29/324.
First, determine the total possible ways in which the dice may be rolled:
Since each die contains six faces, there are 6 x 6 x 6 x 6 = 1,296 different possibilities.
Now, we must determine how many of these result in a different symbol on each die.
The chart below shows all 24 possible permutations of one symbol on each die,
and in how many ways that you can obtain it:
|
Dice #1
|
Dice #2
|
Dice #3
|
Dice #4
|
# of Ways to get this combination
|
| O |
B |
C |
$ |
2 x 2 x 3 x 1 = 12 |
| O |
B |
$ |
C |
2 x 2 x 1 x 3 = 12 |
| O |
C |
B |
$ |
2 x 1 x 1 x 1 = 2 |
| O |
C |
$ |
B |
2 x 1 x 1 x 1 = 2 |
| O |
$ |
B |
C |
2 x 1 x 1 x 3 = 6 |
| O |
$ |
C |
B |
2 x 1 x 3 x 1 = 6 |
| B |
O |
C |
$ |
2 x 2 x 3 x 1 = 12 |
| B |
O |
$ |
C |
2 x 2 x 1 x 3 = 12 |
| B |
C |
O |
$ |
2 x 1 x 1 x 1 = 2 |
| B |
C |
$ |
O |
2 x 1 x 1 x 1 = 2 |
| B |
$ |
O |
C |
2 x 1 x 1 x 3 = 6 |
| B |
$ |
C |
O |
2 x 1 x 3 x 1 = 6 |
| C |
O |
B |
$ |
1 x 2 x 1 x 1 = 2 |
| C |
O |
$ |
B |
1 x 2 x 1 x 1 = 2 |
| C |
B |
O |
$ |
1 x 2 x 1 x 1 = 2 |
| C |
B |
$ |
O |
1 x 2 x 1 x 1 = 2 |
| C |
$ |
O |
B |
1 x 1 x 1 x 1 = 1 |
| C |
$ |
B |
O |
1 x 1 x 1 x 1 = 1 |
| $ |
O |
B |
C |
1 x 2 x 1 x 3 = 6 |
| $ |
O |
C |
B |
1 x 2 x 3 x 1 = 6 |
| $ |
B |
O |
C |
1 x 2 x 1 x 3 = 6 |
| $ |
B |
C |
O |
1 x 2 x 3 x 1 = 6 |
| $ |
C |
O |
B |
1 x 1 x 1 x 1 = 1 |
| $ |
C |
B |
O |
1 x 1 x 1 x 1 = 1 |
There are 116 ways in which you can roll a different symbol on each die.
So, the probability of getting one different symbol on each die is 116 / 1296, which reduces to 29 / 324.
This is equivalent to a probability of 0.0895 or 8.95%.
James Alarie sent in a list of all 116 ways that this can occur:
1. Orange Bell Dollar Cherry
2. Orange Bell Dollar Cherry
3. Orange Bell Dollar Cherry
4. Orange Bell Cherry Dollar
5. Orange Bell Cherry Dollar
6. Orange Bell Cherry Dollar
7. Orange Bell Dollar Cherry
8. Orange Bell Dollar Cherry
9. Orange Bell Dollar Cherry
10. Orange Bell Cherry Dollar
11. Orange Bell Cherry Dollar
12. Orange Bell Cherry Dollar
13. Orange Cherry Bell Dollar
14. Orange Cherry Dollar Bell
15. Orange Dollar Bell Cherry
16. Orange Dollar Bell Cherry
17. Orange Dollar Bell Cherry
18. Orange Dollar Cherry Bell
19. Orange Dollar Cherry Bell
20. Orange Dollar Cherry Bell
21. Orange Bell Dollar Cherry
22. Orange Bell Dollar Cherry
23. Orange Bell Dollar Cherry
24. Orange Bell Cherry Dollar
25. Orange Bell Cherry Dollar
26. Orange Bell Cherry Dollar
27. Orange Bell Dollar Cherry
28. Orange Bell Dollar Cherry
29. Orange Bell Dollar Cherry
30. Orange Bell Cherry Dollar
31. Orange Bell Cherry Dollar
32. Orange Bell Cherry Dollar
33. Orange Cherry Bell Dollar
34. Orange Cherry Dollar Bell
35. Orange Dollar Bell Cherry
36. Orange Dollar Bell Cherry
37. Orange Dollar Bell Cherry
38. Orange Dollar Cherry Bell
39. Orange Dollar Cherry Bell
40. Orange Dollar Cherry Bell
41. Bell Orange Dollar Cherry
42. Bell Orange Dollar Cherry
43. Bell Orange Dollar Cherry
44. Bell Orange Cherry Dollar
45. Bell Orange Cherry Dollar
46. Bell Orange Cherry Dollar
47. Bell Orange Dollar Cherry
48. Bell Orange Dollar Cherry
49. Bell Orange Dollar Cherry
50. Bell Orange Cherry Dollar
51. Bell Orange Cherry Dollar
52. Bell Orange Cherry Dollar
53. Bell Cherry Orange Dollar
54. Bell Cherry Dollar Orange
55. Bell Dollar Orange Cherry
56. Bell Dollar Orange Cherry
57. Bell Dollar Orange Cherry
58. Bell Dollar Cherry Orange
59. Bell Dollar Cherry Orange
60. Bell Dollar Cherry Orange
61. Bell Orange Dollar Cherry
62. Bell Orange Dollar Cherry
63. Bell Orange Dollar Cherry
64. Bell Orange Cherry Dollar
65. Bell Orange Cherry Dollar
66. Bell Orange Cherry Dollar
67. Bell Orange Dollar Cherry
68. Bell Orange Dollar Cherry
69. Bell Orange Dollar Cherry
70. Bell Orange Cherry Dollar
71. Bell Orange Cherry Dollar
72. Bell Orange Cherry Dollar
73. Bell Cherry Orange Dollar
74. Bell Cherry Dollar Orange
75. Bell Dollar Orange Cherry
76. Bell Dollar Orange Cherry
77. Bell Dollar Orange Cherry
78. Bell Dollar Cherry Orange
79. Bell Dollar Cherry Orange
80. Bell Dollar Cherry Orange
81. Cherry Orange Bell Dollar
82. Cherry Orange Dollar Bell
83. Cherry Orange Bell Dollar
84. Cherry Orange Dollar Bell
85. Cherry Bell Orange Dollar
86. Cherry Bell Dollar Orange
87. Cherry Bell Orange Dollar
88. Cherry Bell Dollar Orange
89. Cherry Dollar Orange Bell
90. Cherry Dollar Bell Orange
91. Dollar Orange Bell Cherry
92. Dollar Orange Bell Cherry
93. Dollar Orange Bell Cherry
94. Dollar Orange Cherry Bell
95. Dollar Orange Cherry Bell
96. Dollar Orange Cherry Bell
97. Dollar Orange Bell Cherry
98. Dollar Orange Bell Cherry
99. Dollar Orange Bell Cherry
100. Dollar Orange Cherry Bell
101. Dollar Orange Cherry Bell
102. Dollar Orange Cherry Bell
103. Dollar Bell Orange Cherry
104. Dollar Bell Orange Cherry
105. Dollar Bell Orange Cherry
106. Dollar Bell Cherry Orange
107. Dollar Bell Cherry Orange
108. Dollar Bell Cherry Orange
109. Dollar Bell Orange Cherry
110. Dollar Bell Orange Cherry
111. Dollar Bell Orange Cherry
112. Dollar Bell Cherry Orange
113. Dollar Bell Cherry Orange
114. Dollar Bell Cherry Orange
115. Dollar Cherry Orange Bell
116. Dollar Cherry Bell Orange