Probability of Choosing Same Letters or Vowels from Words: ASSESSMENT and EXAMINATION
In this article, we will explore two interesting problems related to probability involving the words ASSESSMENT and EXAMINATION. We will dive into the calculations, step-by-step explanations, and relevant combinations to understand the likelihood of certain outcomes.
Problem 1: ASSESSMENT
How can we calculate the probability of choosing two letters randomly from the word ASSESSMENT such that either both letters chosen are the same or both letters are vowels?
Understanding the Word ASSESSMENT
The word ASSESSMENT contains 10 letters: A, S, S, E, S, S, E, M, E, N, T. Here, we have:
3 vowels: A, E, E, E Vowel characters with multiple appearances: E (3 times) and S (4 times)Step-by-Step Calculation
1. **Total Number of Ways to Choose Two Letters:** - We can choose any two letters from 10 in ( binom{10}{2} 45 ) ways.
2. **Choosing Two Vowels:** - There are 3 vowels (A, E, E, E). We can choose 2 vowels in ( binom{3}{2} 3 ) ways.
3. **Choosing Two Letters of the Same Character:** - For 'S': We can choose 2 letters from 4 in ( binom{4}{2} 6 ) ways. - For 'E': We can choose 2 letters from 3 in ( binom{3}{2} 3 ) ways. - So, the total number of ways to choose letters of the same character (including both 'S' and 'E') is ( 6 3 9 ) ways.
4. **Adjusting for Overlapping Cases:** - We need to subtract the cases where both chosen letters are the same and vowels (i.e., 'EE'). This has already been counted in the 'E' cases. - Therefore, the adjustment is 1 (for 'EE').
5. **Final Calculation:** - The total number of ways to choose two letters such that either both are the same or both are vowels is ( 9 - 1 8 ).
6. **Probability Calculation:** - The probability is then ( frac{8}{45} approx 0.177 ).
Problem 2: EXAMINATION
Let's solve a similar problem for the word EXAMINATION. We need to find the probability that two randomly selected letters are the same.
Understanding the Word EXAMINATION
The word EXAMINATION contains 11 letters: E, X, A, M, I, N, A, T, I, O, N. Here, we have:
Letters that appear more than once: A (2 times), I (2 times), N (2 times)Step-by-Step Calculation
1. **Prime Assumption:** - Selection without replacement is assumed as the question demands a probability calculation. This assumption simplifies the calculation significantly.
2. **Probability Calculation:** - We need to consider two mutually exclusive scenarios: - Probability that the chosen letters are the same but not vowels (A, A, I, I, N, N). - Probability that both letters are vowels (no vowels in EXAMINATION, so this is 0). - For the letters 'A': - Probability of choosing the first 'A' is ( frac{2}{11} ). - Probability of choosing the second 'A' from the remaining 10 cards is ( frac{1}{10} ). - Therefore, the combined probability for 'A' is: [ frac{2}{11} times frac{1}{10} frac{2}{110} frac{1}{55} ] - For the letters 'I': - Probability of choosing the first 'I' is ( frac{2}{11} ). - Probability of choosing the second 'I' from the remaining 10 cards is ( frac{1}{10} ). - Therefore, the combined probability for 'I' is: [ frac{2}{11} times frac{1}{10} frac{2}{110} frac{1}{55} ] - For the letters 'N': - Probability of choosing the first 'N' is ( frac{2}{11} ). - Probability of choosing the second 'N' from the remaining 10 cards is ( frac{1}{10} ). - Therefore, the combined probability for 'N' is: [ frac{2}{11} times frac{1}{10} frac{2}{110} frac{1}{55} ] - Adding the probabilities for 'A', 'I', and 'N' gives: [ frac{1}{55} frac{1}{55} frac{1}{55} frac{3}{55} ]
3. **Conclusion:** - The total probability of choosing two letters that are the same is ( frac{3}{55} ) or approximately 0.0545.
Summary
In this article, we have explored the probability of choosing two letters randomly from the words ASSESSMENT and EXAMINATION such that either both letters are the same or both letters are vowels. Through detailed calculations, we have determined the probability for each scenario, providing a clear understanding of the underlying concepts and practical applications.
Keywords: probability, combination, vowels, ASSESSMENT, EXAMINATION