How can the faces of two cubes be numbered so that when they are rolled, the resulting sum is a number 1 to 12 inclusive and each sum has the same probability?
Well, to achieve that, we should first hire a professional numbers magician. They have a special ability to conjure up the perfect numbering for our cubes. Alternatively, we could ask the local jokester to come up with some hilarious and randomly selected numbers for us. That way, when we roll the cubes, we'll never be too sure what we'll get â except for lots of laughter!
To number the faces of two cubes in a way that each sum from rolling them has the same probability, you can follow these steps:
Step 1: Assign numbers from 1 to 6 on each face of the first cube.
Step 2: Assign numbers from 1 to 6 on each face of the second cube, but add 6 to each number. This means, for example, you would have 7, 8, 9, 10, 11, and 12 on the faces of the second cube.
Step 3: Now, when you roll both cubes, the minimum sum you can get is 1 + 7 = 8, and the maximum sum is 6 + 12 = 18. However, we want the sum to be between 1 and 12 inclusive, so we need to adjust the numbers on the faces of the cubes.
Step 4: To adjust the numbers, subtract 7 from each face on the second cube. This means, for example, you would have 1, 2, 3, 4, 5, and 6 on the faces of the second cube.
Step 5: Now, when you roll both cubes, the minimum sum you can get is 1 + 1 = 2, and the maximum sum is 6 + 6 = 12. All the possible sums from 2 to 12 inclusive have an equal probability of occurring.
Following these steps, you have successfully numbered the faces of two cubes to ensure each sum has the same probability.
To number the faces of two cubes in a way that each sum has the same probability when rolled, you need to ensure that every possible sum from 1 to 12 has an equal number of combinations. There are different ways to achieve this, but one common method is the following:
1. Start by numbering the faces of the first cube with the numbers 1, 2, 3, 4, 5, and 6.
2. Number the faces of the second cube with the numbers 1, 2, 3, 4, 5, and 6 as well.
3. To ensure that the sums of the two cubes are equally likely, you need to arrange the numbers in a way that the different combinations of sums appear with equal frequency.
4. You can achieve this by pairing numbers in a specific way. For example, pair 1 from the first cube with 1, 2, 3, 4, 5, and 6 from the second cube. Then, pair 2 from the first cube with 1, 2, 3, 4, 5, and 6 from the second cube. Continue this pattern for the remaining numbers on the first cube.
5. This numbering scheme ensures that every possible sum from 2 (1 + 1) to 12 (6 + 6) has exactly one combination. For example, to get a sum of 7, you can have either (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), or (6, 1).
6. When both cubes are rolled, each sum has an equal probability of occurring since there is an equal number of combinations for each possible sum.
By using this numbering scheme, you can ensure that the sum of the two cubes has an equal probability from 1 to 12 inclusive.