Calculating Probability: A Letter from 'Method' Not in 'Mathematics'
Introduction to the Problem
In the realm of probability, understanding the likelihood of certain events is crucial. One such intriguing problem arises when we consider the letters in the words 'method' and 'mathematics'. Specifically, we want to determine the probability that a letter chosen at random from the word 'method' does not appear in 'mathematics'. This problem requires a clear understanding of the letters present in each word and a thorough analysis to calculate the required probability.
Identifying the Letters in Each Word
The first step in solving this problem is to identify the unique letters in both words.
Letters in 'method': m, e, t, h, o, d
Letters in 'mathematics': m, a, t, h, e, m, a, t, i, c, s
After removing the repeated letters, we see that the unique letters in 'mathematics' are: m, a, t, h, e, i, c, s. Therefore, the unique letters in 'method' are: m, e, t, h, o, d.
Determining the Relevant Letters
To find the letters that do not appear in 'mathematics', we compare the letters from 'method'. We exclude the letters that are present in 'mathematics': m, a, t, h, e, i, c, s. This leaves us with:
Letters in 'method' not in 'mathematics': o, d
Counting the Total Letters and Those That Do Not Appear
Total letters in 'method': 6 (m, e, t, h, o, d)
Letters that do not appear in 'mathematics': 2 (o, d)
Calculating the Probability
Now, we calculate the probability that a randomly chosen letter from 'method' is one of the two letters ('o' or 'd') that do not appear in 'mathematics'.
The probability is given by:
Probability (frac{text{Number of letters not in mathematics}}{text{Total number of letters in method}}) (frac{2}{6} frac{1}{3})
Therefore, the probability that a letter chosen at random from the word 'method' does not appear in 'mathematics' is (frac{1}{3}).
Understanding the Solution
This solution can be intuitively understood as follows:
From the six letters in 'method', only two letters (o and d) do not appear in 'mathematics'. Therefore, the probability of randomly selecting a letter that does not appear in 'mathematics' is the ratio of the number of such letters to the total number of letters in 'method'. This ratio is 2/6, which simplifies to 1/3.
Conclusion
The probability that a letter chosen at random from the word 'method' does not appear in the word 'mathematics' is (frac{1}{3}). This problem not only reinforces the importance of identifying unique elements in sets but also highlights the practical application of probability in understanding letter frequency and distribution.