Calculating the Probability of Selecting a Vowel from the Word ‘Probability’

Understanding the concept of probability is crucial in various real-world scenarios, from simple word problems to complex statistical analyses. Here, we focus on a specific word problem: determining the probability of selecting a vowel at random from the word 'probability'. This problem serves as an excellent example for exploring the fundamental principles of probability.

Introduction to the Problem

The word 'probability' contains 11 letters, and among these, there are a total of 4 vowels: o, a, i, i, and i. The task is to calculate the probability of randomly selecting a vowel from this word.

Step-by-Step Calculation

To find the probability of selecting a vowel, we use the following formula:

[ text{Probability of selecting a vowel} frac{text{Number of vowels}}{text{Total number of letters}} ]

Counting the Vowels

The word 'probability' consists of 11 letters. Among these, the vowels are:

o (1) a (1) i (2)

Adding these up, we get a total of 4 vowels.

Calculating the Probability

Using the formula:

[ text{Probability of selecting a vowel} frac{4}{11} ]

This fraction is already in its simplest form, as 4 and 11 have no common factors other than 1.

Verification and Considerations

Let's verify the calculation by considering the total number of letters and the number of vowels:

Letter Vowel p F r F o T b F a T b F i T a T b F i T l F

Here, 'T' represents a vowel, and 'F' represents a consonant. The counts are consistent with our earlier calculation.

Advanced Considerations

Considering 'Y' as a Vowel

In some contexts, the letter 'y' can be considered a vowel. If we include 'y' as a vowel, the new set of vowels in 'probability' becomes o, a, i, i, i, and y, resulting in a total of 6 vowels.

Following the same formula:

[ text{Probability of selecting a vowel} frac{6}{11} ]

This fraction is also in its simplest form.

Simultaneous Vowel Selection Probability

Beyond individual selection, we can also consider the probability of selecting two vowels simultaneously under the condition that they are chosen with replacement. This introduces another layer of complexity.

The probability of selecting a vowel both times:

[ (frac{4}{11})^2 frac{16}{121} approx 0.132258 ]

For the probability of selecting two 'i''s:

[ (frac{2}{11})^2 frac{4}{121} approx 0.033113 ]

These calculations offer insights into the likelihood of specific outcomes in repeated trials.

Conclusion

In conclusion, the probability of randomly selecting a vowel from the word 'probability' is (frac{4}{11}), which cannot be further simplified. By understanding this problem, one gains insight into the fundamental principles of probability and how to apply mathematical reasoning to solve word problems. This knowledge can be extended to a variety of practical scenarios in fields such as data analysis, statistics, and even everyday decision-making.