Problem 1 - Algebra 2

Question

Suppose you select 2 letters at random from the word compute. Find each probability. \(P(2 \text { vowels })\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{7}\).

Step 1: Identify Vowels in 'compute'

First, we need to recognize the vowels in the word 'compute'. The vowels are 'o', 'u', and 'e', so there are 3 vowels in total.

Step 2: Calculate Total Ways to Select 2 Letters

The total number of ways to select 2 letters from the 7 letters in 'compute' is calculated using combinations: \[\binom{7}{2} = \frac{7\times6}{2\times1} = 21\]

Step 3: Calculate Ways to Select 2 Vowels

Next, calculate the ways to select 2 vowels from the 3 available vowels using combinations:\[\binom{3}{2} = \frac{3\times2}{2\times1} = 3\]

Step 4: Determine the Probability

The probability of selecting 2 vowels out of the total 2 letters selected is the number of favorable outcomes divided by the total number of possible outcomes: \[P(2 \text{ vowels}) = \frac{3}{21} = \frac{1}{7}\]

Key Concepts

Probability in Combinatorics

Probability is a measure of how likely an event is to occur. In combinatorics, probability helps us determine the likelihood of selecting particular combinations from a set.
For instance, when we randomly choose two letters from the word "compute," probability aids in calculating the chance that both letters are vowels. This is done by considering the number of favorable outcomes, such as selecting vowels, divided by the total number of possible selections.
The formula for probability is given by:

• Probability = \( \frac{ \text{Number of Favorable Outcomes} }{ \text{Total Number of Outcomes} } \)

In the example of choosing 2 vowels from compute, the probability is calculated as \( \frac{3}{21} \), resulting in \( \frac{1}{7} \). This means there is a 1 in 7 chance of picking two vowels at random from the word.

Understanding Combinations

Combinations are a fundamental part of combinatorics, used to find the number of ways to select items from a larger set. Order does not matter in combinations, making them distinct from permutations.
The formula for combinations for selecting \( r \) items from \( n \) is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

In our exercise, to find out how many ways you can choose 2 letters from 7, you calculate \( \binom{7}{2} = 21 \). If we focus only on vowels, \( \binom{3}{2} = 3 \) gives us the number of ways to select 2 vowels from 3. These values are crucial in determining the probability of certain selections, like only getting vowels.

Identifying and Counting Vowels

Vowels are special types of letters in the English language: 'a', 'e', 'i', 'o', 'u', and occasionally 'y'. In the context of words, identifying vowels helps in calculating specific probabilities, such as selecting them over consonants.
In the word "compute," the vowels are 'o', 'u', and 'e'. This amounts to three vowels in the set of seven letters.
When calculating probabilities, understanding which letters are vowels allows us to determine the favorable outcomes more accurately. For instance, when asked to calculate the probability of drawing two vowels from "compute," recognizing that there are three vowels helps form the basis for further calculations and correct results.