Probability of Selecting a Consonant and a Vowel Randomly from the English Alphabet

There are 26 letters in the English alphabet, consisting of 5 vowels and 21 consonants. When two letters are selected randomly, what is the probability that one is a consonant and the other is a vowel? This article explains the calculation step by step, providing a clear understanding of the probability involved.

Introduction to the English Alphabet

The English alphabet comprises 26 letters, with each letter having a specific place in either the category of vowels or consonants. Vowels are the letters that typically represent a sound when pronounced, while consonants are any letters that are not vowels.

Calculating the Probability

To calculate the probability that one selected letter is a consonant and the other is a vowel, we can follow these steps:

Step 1: Identifying the Total Number of Outcomes

The total number of ways to select 2 letters out of 26 is given by the combination formula (26 choose 2).

Step 2: Identifying the Favorable Outcomes

We need to find the number of ways to select one consonant and one vowel. The number of ways to select 1 consonant out of 21 consonants is (21 choose 1), and the number of ways to select 1 vowel out of 5 vowels is (5 choose 1).

The total number of favorable outcomes can be calculated by multiplying these two values:

[text{Favorable outcomes} 21 times 5]

Step 3: Calculating the Probability

The probability is then calculated by dividing the number of favorable outcomes by the total number of outcomes:

[frac{text{Favorable outcomes}}{text{Total outcomes}} frac{21 times 5}{26 choose 2}]

Calculating the combination (26 choose 2):

[26 choose 2 frac{26 times 25}{2 times 1} 325]

Therefore, the probability is:

[frac{21 times 5}{325} frac{105}{325} frac{21}{65}]

What If Y is Considered a Special Case?

The example provided mentions that the probability calculation might vary if Y is considered a special case. This is because Y can act as either a vowel (in words like 'myth' or 'psychology') or a consonant (in words like 'yellow').

If Y is considered a vowel:

[text{Probability} frac{6}{26} frac{3}{13}]

If Y is considered a consonant:

[text{Probability} frac{5}{26}]

In each case, the probability is adjusted accordingly.

Conclusion

By understanding the principles of probability and using the combination formula, we can accurately determine the probability of selecting a consonant and a vowel randomly from the English alphabet. This confirms that the probability of either a vowel or a consonant being selected is a sure event, resulting in a probability of 1.

Additional Considerations

It is important to recognize that the classification of Y as a vowel or consonant can affect the outcome, making it a variable in probability calculations. Nonetheless, the standard probability for a randomly selected letter being a vowel or a consonant remains 1, given the exhaustive nature of the alphabet's composition.

Key Terms: probability, consonant, vowel, English alphabet