Problem 2 - Algebra 2

Question

Suppose you select 2 letters at random from the word compute. Find each probability. P(2 consonants)

Step-by-Step Solution

Answer

The probability of selecting 2 consonants from 'compute' is \( \frac{2}{7} \).

Step 1: Identify the total number of letters

The word 'compute' has a total of 7 letters: C, O, M, P, U, T, and E.

Step 2: Identify consonants in 'compute'

The consonants in 'compute' are C, M, P, and T, making a total of 4 consonants.

Step 3: Calculate the total ways to select 2 letters

We calculate the total number of ways to select 2 letters from 7. This is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. \[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

Step 4: Calculate the ways to select 2 consonants

We calculate the number of ways to choose 2 consonants from the total of 4 consonants using the combination formula.\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

Step 5: Find the probability of selecting 2 consonants

The probability of selecting 2 consonants is the ratio of the number of ways to select 2 consonants to the total number of ways to select 2 letters. \[ P(2 \text{ consonants}) = \frac{\binom{4}{2}}{\binom{7}{2}} = \frac{6}{21} = \frac{2}{7} \]

Key Concepts

Understanding Combinations

When tackling problems involving probability where choices or selections are made, combinations play a key role. Combinations help us determine how many different ways we can select items from a group, without considering the order in which they are selected. This makes them different from permutations, which do consider order.
The general formula for combinations is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]Where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to choose.
• \(!\) denotes factorial, meaning \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).

When we apply this formula, it tells us how many different ways we can select \( r \) items from \( n \) without caring about the order. It is especially useful in situations requiring probability calculations, such as determining the chances of picking certain letters from a word, as seen in the problem involving the word 'compute.'

Consonants Explained

Consonants are the sounds in speech and the letters in writing that aren't vowels. In the English alphabet, vowels are A, E, I, O, and U. All other letters are considered consonants. Understanding which letters are consonants is crucial when solving the exercise related to probability. In the word 'compute', the consonants are:

• C
• M
• P
• T

This means 'compute' has 4 consonants out of 7 total letters. When asked to find the probability of selecting two consonants, we need to focus only on these specific letters.
Knowing the difference between consonants and vowels helps in categorizing letters during selection events and understanding the constraints or conditions in probability-related exercises.

Decoding Letters in Probability Problems

Letters in a word problem, such as the one involving 'compute', serve as the items we are choosing from. Each letter represents a possible choice. When calculating the probability of selecting certain types of letters, we need to use our knowledge of both combinations and the characteristics of each letter, such as whether it's a consonant or vowel.
When posed with a probability question like "What is the probability of selecting 2 consonants from 'compute'?" we:

• Identify the total number of letters.
• Determine the possible subsets of letters that align with the selection criteria (e.g., consonants).
• Apply combinations to find all possible ways of selecting the given number of letters, filtering these based on specified attributes (consonant, vowel).

This methodical approach ensures that we account for all desired characteristics in our selections, thus leading to an accurate probability calculation. It's the blend of mathematical combination techniques and letter classifications that allows us to solve these probability puzzles effectively.