Problem 31 - Algebra 1

Question

WARDROBE Your cousin spills spaghetti sauce on her shirt and asks to borrow a clean shirt from you for the rest of the day. You decide to let her choose from a selection of 4 sweatshirts, 1 hockey shirt, 8 T-shirts, and 3 tank tops. If it is equally likely that your cousin will choose any shirt, what are the odds that she will choose a sweatshirt?

Step-by-Step Solution

Answer

The probability that she will choose a sweatshirt is 0.25 or 25%

Step 1: Count Total Shirts

First, find the total number of shirts available. This is done by adding the number of each type of shirt together. Number of shirts = 4 (sweatshirts) + 1 (hockey shirt) + 8 (T-Shirts) + 3 (tank tops) = 16 shirts.

Step 2: Identify Sweatshirts

Next, identify the number of sweatshirts available which is 4.

Step 3: Calculate Probability

Then, calculate the probability of choosing a sweatshirt. Probability is calculated as the number of favorable outcomes (number of sweatshirts) divided by the total number of outcomes (total number of shirts). So, the probability = 4 (sweatshirts) / 16 (total shirts) = 0.25 or 25%.

Key Concepts

Understanding the Counting Principle

The counting principle is a fundamental concept in probability, essential for determining how many possible outcomes exist for an event or series of events. In practical terms, it allows us to count the number of ways an event can occur. By determining how many shirts are available, we use the counting principle to calculate the total number of shirt combinations possible.To apply the counting principle, list the different types of shirts available:

• 4 Sweatshirts
• 1 Hockey shirt
• 8 T-Shirts
• 3 Tank tops

By adding these numbers, we account for all possible choices your cousin could make. The calculation is simple yet powerful: \[4 + 1 + 8 + 3 = 16\]Thus, there are 16 shirts altogether, representing 16 potential outcomes for her shirt choice. This total is essential for further probability calculations, as it represents the complete set of options from which the selected shirt will be drawn.

Identifying Favorable Outcomes

The term 'favorable outcomes' refers to the specific outcomes within our total set that meet a desired condition. In this scenario, the condition is your cousin choosing a sweatshirt. When exploring probability, it's important to isolate these favorable outcomes because they form the basis of our probability calculation. Let's revisit the available shirts:

• 4 Sweatshirts (favorable outcomes)
• 1 Hockey shirt
• 8 T-Shirts
• 3 Tank tops

Among these, the sweatshirts are our favorable outcomes. The key here is identifying the exact number that matches our desired condition, which is choosing a sweatshirt. In our case, there are 4 sweatshirts available. These 4 instances represent the favorable outcomes because each is a situation where her choice results in her wearing a sweatshirt. Simple counting gives us this crucial figure needed for calculating probability.

Calculating Total Outcomes

Total outcomes encompass all possible results of a random choice or selection. In probability, identifying the total outcomes sets the stage for determining how likely an event is to occur. It includes every single choice your cousin has regarding the type of shirt she might pick. With the total number of shirts tallied earlier using the counting principle, we established there are 16 shirts. These 16 shirts make up our total outcomes:

• 4 Sweatshirts
• 1 Hockey shirt
• 8 T-Shirts
• 3 Tank tops

This comprehensive view accounts for every possibility in the selection process. For probability purposes, having a clear understanding of total outcomes ensures that our comparisons against favorable outcomes are accurate. In a scenario like this, where one shirt is chosen randomly, knowing the total outcomes allows us to frame our probability calculations precisely. Each outcome is equally likely, making the concept of total outcomes a central part of evaluating the odds of her selecting any given type of shirt, such as a sweatshirt.