Question

All eleven letters from the word MATHEMATICS are written on individual slips of paper and placed in a hat. If you reach into the hat and randomly choose one slip of paper, what are the odds against the paper having a vowel written on it? State your answer as a ratio (two numbers separated by a colon), but do not put any spaces in your response.

Answer 1

Given the question in the question tab, the following are the steps to solve the problem

Step 1: State the formula for probability

Sample Space is the number of letters in MATHEMATICS

Expected outcome is the number of letters which are not vowels in MATHEMATICS

`\begin{gathered} P=\frac{no\text{ of expected outcomes}}{Sample\text{ Space}} \\ Sample\text{ Space=11} \\ no\text{ of expected outcomes=}7 \\ \end{gathered}`

Step 2: Calculate the chance of choosing a paper which is not a vowel

`\begin{gathered} P_{\text{vowel}}=1-\frac{n(\text{vowels)}}{n(\text{total)}} \\ P_{\text{not vowel}}=1-(4)/(11)=(7)/(11) \end{gathered}`

Step 3: Express the odds against the paper having a vowel written on it as a ratio

The outcome of 7/11 expressed as ratio of vowels against total outcome will be:

`\begin{gathered} \text{not vowels:vowels} \\ =7\colon(11-7) \\ =7\colon4 \end{gathered}`

Hence, the odds against the paper having a vowel written on it as a ratio is 7:4