Solution: 2017 Winter Final - 10

Author: Michiel Smid

Question

We choose, uniformly at random, a string consisting of 14 characters, where each character is a lowercase letter. Let $A$ be the event

(A vowel is one of the letters $a$, $e$, $i$, $o$, and $u$.) What is $\Pr(A)$?

(a)

$1 - (21/26)^{14}$

(b)

$14 \cdot (5/26) \cdot (21/26)^{13}$

(c)

$5 \cdot (5/26) \cdot (21/26)^{13}$

(d)

$1 - (26/21)^{14}$

Let S be all possible strings of length 14: $ |S| = 26^{14} $

Let B be the event that the string contains at no vowels

The first character can be any of the 21 consonants: 21

The second character can be any of the 21 consonants: 21

…

The 14th character can be any of the 21 consonants: 21

$ |B| = 21^{14} $

$ Pr(B) = \frac{21^{14}}{26^{14}} $

$ Pr(A) = 1 - Pr(B) $

$ Pr(A) = 1 - \frac{21^{14}}{26^{14}} $