Problem 13 - Algebra 2 and Trigonometry

Question

The letters of the word STRAWBERRIES are written on cards and the cards are then shuffled. A card is picked at random. Find the probability that the card contains: $$ \begin{array}{ll}{\text { a. the letter } S} & {\text { b. the letter } R} \\\ {\text { c. a consonant }} & {\text { d. a letter in the word CREAM }}\end{array} $$

Step-by-Step Solution

Answer

a. \( \frac{1}{6} \), b. \( \frac{1}{4} \), c. \( \frac{2}{3} \), d. \( \frac{1}{2} \).

Step 1: Count Total Letters

First, determine the total number of letters in the word STRAWBERRIES. Count each letter: S, T, R, A, W, B, E, R, R, I, E, S. There are 12 letters in total.

Step 2: Calculate Probability of Picking 'S'

Count the number of 'S' letters in the word STRAWBERRIES. There are 2 S's in the word. Use the formula for probability: \( P(S) = \frac{\text{Number of 'S' cards}}{\text{Total cards}} = \frac{2}{12} = \frac{1}{6} \).

Step 3: Calculate Probability of Picking 'R'

Count the number of 'R' letters in STRAWBERRIES. There are 3 R's. So, the probability is \( P(R) = \frac{3}{12} = \frac{1}{4} \).

Step 4: Calculate Probability of Picking a Consonant

Identify consonants in the word: S, T, R, W, B, R, R, S. There are 8 consonants. Therefore, the probability is \( P(\text{consonant}) = \frac{8}{12} = \frac{2}{3} \).

Step 5: Calculate Probability of Picking a Letter in "CREAM"

Identify if each letter in 'CREAM' is present in 'STRAWBERRIES': C (0), R (3), E (2), A (1), M (0). The letters R, E, and A appear, making 6 matches in total (3 R's, 2 E's, 1 A). Thus, the probability is \( P(\text{letter in CREAM}) = \frac{6}{12} = \frac{1}{2} \).

Key Concepts

Counting

Counting is a fundamental concept in probability. It involves identifying and tallying the elements in a set that we are interested in. Let's take the word STRAWBERRIES as an example.
This word consists of 12 letters in total. To solve problems related to probability in this context, the first step is to count the total number of cards, or letters, that we have. This sets the stage for evaluating probability. When we talk about counting in probability, it usually means we need to:

• Count the total number of items.
• Identify and count the specific items of interest (like the 'S' or 'R').

Understanding counting is vital because it directly affects the probability calculation. Without knowing the counts, determining the likelihood of an event becomes challenging.

Consonants

In the context of this problem, consonants are all the letters that are not vowels. This includes S, T, R, W, and B in the word STRAWBERRIES. When solving problems involving consonants, our task is to identify these specific letters and count them in the word.
For STRAWBERRIES, there are 8 consonants. Identifying these includes a quick check on which letters in the word are consonants versus vowels. The vowels in this word are A, E, and I. Every other letter is a consonant.
Calculating the probability of picking a consonant involves:

• Counting the number of consonants.
• Using the total number of letters to find the probability of selecting one of these consonants when picking a card at random.

This forms the basis for calculating events involving consonants in probability scenarios.

Alphabetical Probability

Alphabetical probability deals with the likelihood of randomly selecting certain letters from an alphabet in a given set. In this problem, we focus on finding the probability of picking specific letters such as 'S', 'R', or any consonant from STRAWBERRIES.
To compute these probabilities, use the formula:
\[ P( ext{event}) = \frac{ ext{Number of favorable outcomes}}{ ext{Total possible outcomes}} \]
For example, to find the probability of picking an 'R':

• Count the number of 'R's in the word (3).
• Divide by the total number of letters (12).

Thus, the probability is \( \frac{3}{12} = \frac{1}{4} \). This simple method allows us to calculate the probability for any single letter or combination of letters.

Step-by-Step Solutions

A step-by-step solution is a guided process that helps in understanding how to arrive at an answer logically and effectively. Let's break down the approach with the example of letters from STRAWBERRIES.
In a step-by-step solution, each action is simplified to guide you through the process:

• Step 1: Count the total items, which in this case, are the letters (12 total).
• Step 2: Determine the number of specific letters ('S', 'R', consonants) and use these counts to calculate probabilities.
• Step 3: Calculate probabilities for compound events, such as letters found in another word (CREAM), by aligning the count of matching letters.

Each calculation uses basic probability principles by dividing the number of times an event occurs by the total number of events. Using this systematic approach makes solving probability problems structured and understandable.