To solve the problem, we first need to identify the components of the word "replacement" in terms of letters and vowels.
The word "replacement" has 11 letters in total. To categorize them:
-
Counting the letters:
- r, e, p, l, a, c, e, m, e, n, t
- Total letters: 11
-
Counting the vowels:
- The vowels in the word are: e, a, e, e (3 e's and 1 a). Thus, we have:
- Vowels: e (3), a (1) → Total vowels = 4
-
Counting the consonants:
- The consonants in the word are: r, p, l, c, m, n, t → Total consonants = 7
Now we calculate the probability of first selecting a vowel and then selecting a consonant (not a vowel).
Step 1: Probability of selecting a vowel first
The probability of selecting a vowel first: \[ P(\text{Vowel first}) = \frac{\text{Number of vowels}}{\text{Total number of letters}} = \frac{4}{11} \]
Step 2: Probability of selecting a consonant second
After selecting a vowel first, there will now be 10 letters left (since we do not replace the first card), and out of these 10 letters, 7 are consonants.
The probability of now selecting a consonant after selecting a vowel: \[ P(\text{Consonant second | Vowel first}) = \frac{\text{Number of consonants left}}{\text{Total number of letters left}} = \frac{7}{10} \]
Step 3: Combined probability
To find the overall probability of both events happening (selecting a vowel first, followed by a consonant), we multiply the probabilities of the two independent events: \[ P(\text{Vowel first, then Consonant}) = P(\text{Vowel first}) \times P(\text{Consonant second | Vowel first}) = \frac{4}{11} \times \frac{7}{10} \]
Calculating that gives: \[ P(\text{Vowel first, then Consonant}) = \frac{4 \times 7}{11 \times 10} = \frac{28}{110} = \frac{14}{55} \]
Thus, the probability of choosing a vowel and then not a vowel is: \[ \boxed{\frac{14}{55}} \]