Each of the letters of the word 'BINOCULARS' is written on identical cards and put in a bag. They are well shuffled. If a card is drawn at random from the bag, find the probability that the letter is:
(a) a vowel
(b) one of the first 8 letters of the English alphabet which appears in the given word.
(Support your working with at least one sample space)
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The word 'BINOCULARS' has 10 letters, all distinct.
Sample space = {B, I, N, O, C, U, L, A, R, S}
Total number of outcomes = 10.
(a) The vowels in the word are I, O, U, A.
Number of vowels = 4.
By formula,
Probability = $\frac{\text{No. of favourable outcomes}}{\text{Total number of outcomes}}$
$\Rightarrow P(\text{vowel}) = \frac{4}{10} = \frac{2}{5}.$
Hence, the probability that the letter is a vowel = $\mathbf{\frac{2}{5}}$.
(b) The first 8 letters of the English alphabet are A, B, C, D, E, F, G, H.
The letters among these which appear in 'BINOCULARS' are B, C and A.
Number of favourable outcomes = 3.
$\Rightarrow P = \frac{3}{10}.$
Hence, the required probability = $\mathbf{\frac{3}{10}}$.