Problem 3 - Algebra 2

Question

Suppose you select 2 letters at random from the word compute. Find each probability. P(1 vowel, 1 consonant)

Step-by-Step Solution

Answer

The probability is \(\frac{4}{7}\).

Step 1: Identify Vowels and Consonants

The word "compute" has the letters {c, o, m, p, u, t, e}. Among these, the vowels are {o, u, e} and the consonants are {c, m, p, t}. This gives us 3 vowels and 4 consonants.

Step 2: Determine Total Number of Ways to Choose 2 Letters

Calculate the total number of ways to choose 2 letters from the 7 letters in "compute". This is given by the combination formula \(\binom{n}{k}\), so we have \(\binom{7}{2} = 21\).

Step 3: Calculate Ways to Choose 1 Vowel and 1 Consonant

To find the number of ways to choose 1 vowel and 1 consonant, determine the combinations separately. There are 3 ways to choose 1 vowel from 3 and 4 ways to choose 1 consonant from 4. So, we have \(\binom{3}{1} \times \binom{4}{1} = 3 \times 4 = 12\) ways.

Step 4: Calculate Probability

The probability of selecting 1 vowel and 1 consonant is the ratio of favorable outcomes to the total outcomes. Thus, \( P(\text{1 vowel, 1 consonant}) = \frac{12}{21} = \frac{4}{7} \).

Key Concepts

Combinations

In probability, combinations are an essential concept. They help us to determine possible selections from a set without regard to order. When you choose students for a team or, in this exercise, letters from a word, you'd use combinations.

A combination is calculated using the formula \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. For instance, in the word "compute," there are 7 letters. If we want to choose 2 letters at random, we use \( \binom{7}{2} \). This equals 21, meaning there are 21 different ways to choose any 2 letters from "compute."

Combinations are different from permutations, where the order does matter. With combinations, "ab" is the same as "ba." Thus, they are useful when only the selection itself is important.

Vowels and Consonants

Understanding the distinction between vowels and consonants is crucial in linguistics and many probability exercises. Vowels are the letters \( \{ a, e, i, o, u \} \), which are pronounced with an open vocal tract. In contrast, consonants are all other letters which involve some constriction or closure in the vocal tract.

In the word "compute," we find vowels \( \{ o, u, e \} \) and consonants \( \{ c, m, p, t \} \). Properly identifying these is the first step when your task involves differentiating between them, like in this problem where we calculate the probability of selecting 1 vowel and 1 consonant.

This identification allows us to find that we have 3 vowels and 4 consonants, from which the selection needs to be made. This foundational understanding is essential to properly set up probability calculations.

Basic Probability

Probability helps us measure how likely an event is to occur. It's a ratio between the favorable outcomes and all possible outcomes. The formula is:

• Probability (P) = Number of Favorable Outcomes / Total Number of Outcomes

In our exercise, we need the probability of choosing 1 vowel and 1 consonant from the word "compute." We already determined there are 12 ways to choose 1 vowel and 1 consonant (favorable outcomes), and from earlier steps, there are 21 total ways to choose any 2 letters.

Thus, the probability of selecting 1 vowel and 1 consonant is \( P = \frac{12}{21} \). Simplifying this fraction gives \( \frac{4}{7} \), so the probability of drawing 1 vowel and 1 consonant from "compute" is \( \frac{4}{7} \). Understanding these basic calculations can help in a wide range of probability problems, making them a critical part of mathematical education.