Each of the letters of the word 'AUTHORIZES' is written on identical circular discs and put in a bag. They are well shuffled. If a disc is drawn at random from the bag, what is the probability that the letter is:
(i) a vowel
(ii) one of the first 9 letters of the English alphabet which appears in the given word.
(iii) one of the last 9 letters of the English alphabet which appears in the given word ?

Answer

To solve the problem, we will break it down into three parts as specified in the question.

### Given:
The word is "AUTHORIZES".

### Total Letters:
The total number of letters in "AUTHORIZES" is 10.

### (i) Probability that the letter is a vowel:
1. **Identify the vowels in the word**: The vowels in "AUTHORIZES" are A, U, O, I, and E.
2. **Count the number of vowels**: There are 5 vowels (A, U, O, I, E).
3. **Calculate the probability**:
\[ P(\text{vowel}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{10} = \frac{1}{2} \]

### (ii) Probability that the letter is one of the first 9 letters of the English alphabet:
1. **Identify the first 9 letters**: The first 9 letters are A, B, C, D, E, F, G, H, I.
2. **Identify which of these letters are in "AUTHORIZES"**: The letters present are A, E, H, and I.
3. **Count the number of favorable outcomes**: There are 4 letters (A, E, H, I).
4. **Calculate the probability**:
\[ P(\text{first 9 letters}) = \frac{4}{10} = \frac{2}{5} \]

### (iii) Probability that the letter is one of the last 9 letters of the English alphabet:
1. **Identify the last 9 letters**: The last 9 letters are R, S, T, U, V, W, X, Y, Z.
2. **Identify which of these letters are in "AUTHORIZES"**: The letters present are R, S, T, U, and Z.
3. **Count the number of favorable outcomes**: There are 5 letters (R, S, T, U, Z).
4. **Calculate the probability**:
\[ P(\text{last 9 letters}) = \frac{5}{10} = \frac{1}{2} \]

### Final Answers:
- (i) Probability of drawing a vowel: \(\frac{1}{2}\)
- (ii) Probability of drawing one of the first 9 letters: \(\frac{2}{5}\)
- (iii) Probability of drawing one of the last 9 letters: \(\frac{1}{2}\)