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If Wayne considers the letter "Y" to be a vowel but Kristen does not, thinking that there are only 5 vowels, by what percent is the probability that a randomly selected letter out of the 26 letter alphabet will be a vowel greater in Wayne's opinion than in Kristen's opinion? A)$5$% ; B) $6$% ; C) $20$% ; D) $30$% ; E) $32$%

So for Wayne, the probability of a randomly selected letter being a vowel is $\frac{6}{26}$. For Kristen, it is $\frac{5}{26}$. Then clearly, the difference is $1/26$, and so I thought the percentage would be $\frac{1}{26} \cdot 100 = 3.85$%, but that's none of the answer choices. Rather, the solution says that I should do $\frac{1/26}{5/26} = 20$%, but I don't understand why $5/26$ should be in the denominator.

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  • $\begingroup$ Where is this question from? $\endgroup$ – Badr B Jun 6 '18 at 2:29
  • $\begingroup$ Whenever we talk about percentages, it is crucial to know what the percentages are of and to have this information communicated. Here, the solution is implying that we are talking about percentages of the probability that kristen thought it should be. $\endgroup$ – JMoravitz Jun 6 '18 at 2:31
  • $\begingroup$ Your answer of $\frac{1}{26}$ is correct if we are talking about percentages of the whole, i.e. percentages of the number of times we run the experiment divided by the number of times we ran the experiment. The book's solution is also correct but is talking about percentages of the number of times kristen thought the randomly selected letter was a vowel divided by the number of times we ran the experiment. $\endgroup$ – JMoravitz Jun 6 '18 at 2:35
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The question statement is phrased somewhat confusingly. Here's a better way to put things :

By what percentage is , "the probability that a randomly selected letter out of the $26$ alphabets will be a vowel in Wayne's opinion" , greater than , "the probability that a randomly selected letter out of the $26$ alphabets will be a vowel in Kristen's opinion".

In other words, by what percentage is "$a$" greater than "$b$"? (Where $a$ and $b$ are something that you need to calculate).

To solve such a question, you find what $a$ and $b$ are, then find out by how much $a$ exceeds $b$, and the ratio between this difference and $b$ is what you are seeking. For example : "by what percentage does $35$ exceed $25$"? Your answer would be $\frac {35 - 25}{25} = \frac 25 = 40%$.

In a similar manner, for the question given to you, we have $a = \frac 6{26}$ and $b = \frac 5{26}$. Now you see why the answer is $\frac 15 = 20%$.

The rephrasing of the question, and therefore understanding it, is sometimes more crucial than the question itself. Next time, you should read questions like this more carefully.

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"by what percent is the probability off" is the key phrase. The probability is $\frac{5}{26}$ and Wayne is off by $\frac{1}{26}$ so as a percentage that's $1/5 \cdot 100\% = 20\%$

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  • $\begingroup$ Um I don't see "off" in the given question statement... $\endgroup$ – space Jun 6 '18 at 3:52
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    $\begingroup$ I simplified it, I used "off" instead of an overly wordy description where "off" equals "a vowel greater". $\endgroup$ – Phil H Jun 6 '18 at 3:58

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