Understanding the Concept
Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. In problems involving selection, we use combinations, denoted as $C(n, r)$ or $\binom{n}{r}$, which represents the number of ways to choose $r$ items from a pool of $n$ items without regard to order.
Analyzing the Word 'EXAMINATION'
First, let's list the letters in the word "EXAMINATION":
- E, X, A, M, I, N, A, T, I, O, N
There are 11 letters in total. Let's count the frequency of each:
- E: 1
- X: 1
- A: 2
- M: 1
- I: 2
- N: 2
- T: 1
- O: 1
Step-by-Step Solution
1. Calculate Total Possible Outcomes
We need to select 2 letters out of 11. The number of ways to do this is: $\binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55$
2. Calculate Favorable Outcomes
We want to select two letters that are the same. This can only happen if we pick a pair of 'A's, 'I's, or 'N's.
- Ways to pick 2 'A's: $\binom{2}{2} = 1$
- Ways to pick 2 'I's: $\binom{2}{2} = 1$
- Ways to pick 2 'N's: $\binom{2}{2} = 1$
Total favorable outcomes = $1 + 1 + 1 = 3$.
3. Calculate Probability
Probability $P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$ $P = \frac{3}{55}$
Conclusion
The probability that both letters selected are the same is $\frac{3}{55}$.