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
Correctly solved by:
| 1. James Alarie | Flint, Michigan |
| 2. Richard O'Leary |
John Paul II Catholic High School, Tallahassee Florida |