Probability of Selecting Two Same Letters from the Word 'Television'

The word television has a collection of letters that can be analyzed for various probability questions. One such question is the likelihood of selecting two letters at random that are the same. In this article, we will explore the steps and calculations to determine this probability.

Introduction to the Word 'Television'

The word 'television' consists of 10 letters: t, e, l, e, v, i, s, i, o, n. By looking at the composition of each letter, we can break down the word into vowels and consonants.

Composition of the Word

The word 'television' has the following letters:

Letters: t, e, l, v, i, s, o, n Vowels: e, e, i, i, o Consonants: t, l, v, s, n

Counting the occurrences of each letter, we have:

t: 1 e: 2 l: 1 v: 1 i: 2 s: 1 o: 1 n: 1

Calculating Total Combinations

First, we calculate the total number of ways to choose 2 letters from 10. This can be done using the combination formula:

(binom{10}{2} frac{10 times 9}{2 times 1} 45)

Favorable Outcomes

The favorable outcomes are the scenarios where the two selected letters are the same. From the word 'television', the letters that appear more than once are 'e' and 'i'. We calculate the number of ways to select 2 of these identical letters:

Ways to select 2 'e's: (binom{2}{2} 1) Ways to select 2 'i's: (binom{2}{2} 1)

Probability Calculation

The probability of selecting two same letters is the ratio of favorable outcomes to the total number of outcomes. We have:

(frac{1 1}{45} frac{2}{45})

Therefore, the probability that the two letters selected at random from the word 'television' are the same is:

(frac{2}{45})

Event-Oriented Analysis

To further understand the probability, let's analyze the two key events:

Event 1: The occurrence of two same letters. Event 2: The occurrence of two different letters.

Event 1: Two Same Letters

As previously calculated, the probability of selecting two 'e's or two 'i's is:

(frac{1}{45}) for each.

Since there are two such cases (two 'e's and two 'i's), we adjust the probability as:

(frac{1}{45} frac{1}{45} frac{2}{45})

Event 2: Two Different Letters

First, we exclude the two identical letters 'e' and 'i', leaving us with 8 different letters. The number of ways to choose 2 letters from 8:

(binom{8}{2} frac{8 times 7}{2 times 1} 28)

Thus, the probability of selecting two different letters is:

(frac{28}{45})

Conclusion

We have also considered the scenarios where both selected letters are vowels or consonants. For vowels:

(binom{5}{2} frac{5 times 4}{2 times 1} 10) so, (frac{10}{45} frac{2}{9}).

Similarly, for consonants, the probability is also:

(frac{10}{45} frac{2}{9}).

Overall, the probability of selecting two same letters from the word 'television' is:

(frac{2}{45}).

Summary of Key Points

Total number of letters: 10 Favorable outcomes (same letters): 2 (e and i) Total possible outcomes: 45 Probability of two same letters: (frac{2}{45})