Probability of Choosing Two Consonants from the Word STUPID

In this article, we will explore the probability of selecting two consonants from the word 'STUPID' when two letters are chosen at random. The word 'STUPID' contains six letters, and we will delve into how to calculate the probability that both letters chosen are consonants.

The word 'STUPID' consists of the following letters: S, T, U, P, I, D. The vowels in this set are U and I. Therefore, the consonants are S, T, P, and D.

Counting the Letters

First, we need to identify the total number of letters and the number of vowels and consonants:

Total letters: 6 (S, T, U, P, I, D) Total vowels: 2 (U, I) Total consonants: 4 (S, T, P, D)

Calculating the Total Number of Combinations

The total number of ways to choose 2 letters from 6 letters is given by the combination formula:

$$ binom{6}{2} frac{6!}{2!(6-2)!} frac{6 times 5}{2 times 1} 15 $$

Next, we calculate the number of ways to choose 2 consonants from the 4 consonants:

$$ binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6 $$

Calculating the Required Probability

The probability that none of the letters chosen are vowels (i.e., both are consonants) is given by the ratio of the number of ways to choose 2 consonants from 4 to the total number of ways to choose 2 letters from 6:

$$ text{Probability} frac{binom{4}{2}}{binom{6}{2}} frac{6}{15} frac{2}{5} 0.4 $$

Alternative Approach Using Inclusion-Exclusion Principle

We can also approach this problem using the inclusion-exclusion principle. Let us consider the total number of two-letter combinations and subtract the combinations that include vowels:

Total number of two-letter combinations from 6 letters: 15 Number of combinations including the vowels: UI can be paired with each of the remaining 4 consonants, giving 4 combinations. Other pairs involving U (but not with I twice), such as US, UT, UP, giving 3 more combinations. Do the same for I, giving another 3 combinations (IT, IP, ID). However, we have counted UI twice, so we need to subtract one combination.

The total number of combinations that include a vowel is 4 (for UI) 3 (for U with consonants) 3 (for I with consonants) - 1 (subtracting the repeated UI) 9.

Therefore, the number of combinations that include only consonants is 15 - 9 6.

The probability that both chosen letters are consonants is:

$$ frac{6}{15} frac{2}{5} 0.4 $$

Conclusion

In summary, the probability that none of the letters chosen are vowels, when two letters are randomly selected from the word 'STUPID', is

$frac{2}{5}$p>. This can be calculated either through the direct combination method or by using the inclusion-exclusion principle.