Rolling a Letter: Probability and Combinatorial Logic
Imagine having a cube with letters abcdef on its faces. When you roll this cube, what is the probability of rolling a letter that is part of the word 'mathematics' or a letter from the name 'Marie'? The answer lies in the art of combinatorial logic, and in this article, we will explore the solution step-by-step.
Understanding the Problem
To solve this problem, let's break it down into smaller parts:
1. Sample Space
Our sample space (S) consists of the letters on the cube: {a, b, c, d, e, f}.
2. Letters in 'Mathematics'
Identify the unique letters in the word 'mathematics': {m, a, t, h, e, i, c, s}.
From the cube, the relevant letters are: {a, e}.
3. Letters in 'Marie'
Identify the unique letters in the name 'Marie': {m, a, r, i, e}.
From the cube, the relevant letters are: {a, e}.
4. Combining the Letters
The letters that are either in 'a' or 'e' are: {a, e}.
5. Counting the Outcomes
The total number of favorable outcomes rolling either 'a' or 'e' is 2. The total number of possible outcomes when rolling the cube is 6, since there are 6 faces.
6. Calculating the Probability
The probability of rolling a letter that is in the word 'mathematics' or a letter from the name 'Marie' is:
(P frac{text{Number of favorable outcomes}}{text{Total number of outcomes}} frac{2}{6} frac{1}{3})
Thus, the probability of rolling a letter that is in the word 'mathematics' or a letter from the name 'Marie' is (frac{1}{3}).
Further Explanation
The sample space for 'Marie' is M {m, a, r, i, e}, and the sample space for 'Mathematics' is T {m, a, t, h, e, m, a, t, i, c, s}.
The common letters in 'Marie' and 'Mathematics' are MT {m, a, i, e}.
However, the letters common to MT and S are only 'a' and 'e', as the cube does not include 'm', 'i', or 'c'.
Therefore, the probability of rolling a letter that is in the word 'Marie' or 'Mathematics' is 2/6 or (frac{1}{3}).
Conclusion
By applying combinatorial logic and understanding the sample space, we can effectively determine the probability of rolling a specific letter on a cube. This problem not only tests basic probability concepts but also highlights the importance of careful enumeration.
Related Keywords
Probability Combinatorial Logic Cube RollingFurther Reading
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