Each of the letters of the word 'AUTHORIZES' is written on identical circular discs and put in a bag. 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 ?

(i) No. of vowels in the word 'AUTHORIZES' = 5 [A, E, I, O, U]

Total no. of letters in the word 'AUTHORIZES' = 10.

P(that disc drawn has a vowel) = $\dfrac{\text{No. of vowels in 'AUTHORIZES'}}{\text{No. of letters in 'AUTHORIZES'}} = \dfrac{5}{10} = \dfrac{1}{2}$.

Hence, P(that disc drawn has a vowel) = $\dfrac{1}{2}$.

(ii) One of the first 9 letters of the English alphabet which appears in the given word are A, E, I, H.

Total no. of letters in the word 'AUTHORIZES' = 10.

Required probability = $\dfrac{\text{No. of first 9 letters in 'AUTHORIZES'}}{\text{No. of letters in 'AUTHORIZES'}} = \dfrac{4}{10} = \dfrac{2}{5}$.

Hence, required probability = $\dfrac{2}{5}$.

(iii) One of the last 9 letters of the English alphabet which appears in the given word are R, S, T, U, Z.

Total no. of letters in the word 'AUTHORIZES' = 10.

Required probability = $\dfrac{\text{No. of last 9 letters in 'AUTHORIZES'}}{\text{No. of letters in 'AUTHORIZES'}} = \dfrac{5}{10} = \dfrac{1}{2}$.

Hence, required probability = $\dfrac{1}{2}$.