Problem 17 - Algebra and Trigonometry

Question

A letter is chosen at random from the word EXTRATERRESTRIAL. Find the probability of the given event. (a) The letter \(T\) is chosen. (b) The letter chosen is a vowel. (c) The letter chosen is a consonant.

Step-by-Step Solution

Answer

(a) \(\frac{3}{16}\); (b) \(\frac{5}{16}\); (c) \(\frac{11}{16}\).

Step 1: Count the total number of letters

The word "EXTRATERRESTRIAL" contains 16 letters. We will use this total to calculate probabilities for the events.

Step 2: Count the number of T's

In the word "EXTRATERRESTRIAL," there are 3 T's. We need this to find the probability of choosing a T.

Step 3: Calculate the probability of choosing a T

The probability of choosing a T is the number of T's divided by the total number of letters: \( P(T) = \frac{3}{16} \).

Step 4: Identify the vowels in the word

The vowels in the word "EXTRATERRESTRIAL" are E, A, and I. Total vowels are: 3 E's, 1 A, 1 I = 5 vowels.

Step 5: Calculate the probability of choosing a vowel

The probability of choosing a vowel is the number of vowels divided by the total number of letters: \( P(Vowel) = \frac{5}{16} \).

Step 6: Count the number of consonants

Consonants are all letters that are not vowels: X, T, R, T, R, S, T, R, L. There are 11 consonants.

Step 7: Calculate the probability of choosing a consonant

The probability of choosing a consonant is the number of consonants divided by the total number of letters: \( P(Consonant) = \frac{11}{16} \).

Key Concepts

Letters in English

In the English alphabet, there are 26 letters. These letters are categorized into vowels and consonants, forming the building blocks of words and sentences. Each letter can be used in various combinations to create words, significantly contributing to language and communication. In exercises dealing with probability and letters, like choosing a random letter from a word, each letter's unique identity plays a crucial role. For example, in the word "EXTRATERRESTRIAL," each letter occurs a certain number of times, contributing to how we calculate probabilities.

Vowels and Consonants

In English, vowels are the life of syllables, forming the core sounds in spoken language. There are 5 main vowels: A, E, I, O, U, although sometimes Y can also act as a vowel. On the other hand, consonants are the speech sounds that involve obstruction of airflow. In most English words, consonants surround the vowels, giving them structure and rhythm. In our exercise word "EXTRATERRESTRIAL," the vowels are E, A, and I. Identifying these vowels helps us to easily calculate the probability of vowel selection. Likewise, consonants in the word include other letters like X, T, R, S, and L, forming a majority, thus affecting their probability outcomes.

Counting Techniques

Counting is a fundamental skill in mathematics that helps us determine the number of outcomes in a given set, which is particularly useful in probability exercises. When working with letters in a word, we often count how many times each letter appears to find the probability of choosing a particular letter.

• First, count the total letters to establish a denominator for probability fractions.
• Next, count occurrences of the letter of interest, like T, or groups like vowels, and use these numbers as numerators.
• For example, in "EXTRATERRESTRIAL," there are 16 letters in total. This total helps calculate probabilities, such as choosing a vowel or consonant. The probabilities are calculated by the formula: \( P(Event) = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} \).

Understanding these counting methods ensures accurate probability calculations, allowing us to analyze other letters or scenarios using similar techniques.