Problem 67 - Algebra 1

Question

Find the probability of the event. Choosing a blue marble from a bag that contains 16 blue marbles and 14 white marbles.

Step-by-Step Solution

Answer

The probability is \( \frac{8}{15} \).

Step 1: Identify the total number of outcomes

To start off, you have to understand the full scope of possible results, which in this context is the total number of marbles in the bag. That is, you must add together 16 blue marbles and 14 white marbles, which equals 30 total marbles.

Step 2: Identify the desired outcome

Next, it's important to pinpoint the specific event you're interested in. In this case, it's picking out a blue marble, and there are 16 blue marbles in the bag.

Step 3: Calculate the probability

The probability is found by dividing the number of desired outcomes by the total number of outcomes. Thus, the probability of drawing a blue marble is \( \frac{16}{30} \). Simplifying the fraction, you get \( \frac{8}{15} \) as the solution.

Key Concepts

Probability Calculation

Understanding the probability of an event involves calculating how likely it is to occur. To do this, one must employ a basic probability formula: the ratio of the number of desired outcomes to the total number of possible outcomes. This ratio gives a number between 0 and 1, often expressed as a percentage, fraction, or decimal. For example, if you want to find the chance of picking a blue marble from a bag, and you have a total of 30 marbles with 16 being blue, the calculation would be \( \frac{16}{30} \), which simplifies to \( \frac{8}{15} \). This forms the crux of probability calculation; it is a fundamental concept that sets the groundwork for more complex probability theories and applications.

In educational settings, making probability calculation understandable is crucial. By breaking down events into their basic elements – desired and total outcomes – students can approach probability problems methodically, minimizing confusion. Visual aids such as diagrams or counters can also be beneficial in illustrating these concepts clearly to enhance comprehension.

Desired Outcome

The desired outcome in a probability context refers to the specific event or result that you're interested in finding the likelihood of. If you're dealing with a deck of cards, your desired outcome might be drawing an ace. In a dice game, it could be rolling a six. It's essential to clearly define what your desired outcome is before you begin calculating probabilities, as this determines the numerator in your probability fraction.

To make this concept easier for students to understand, it is advisable to use relatable examples. Using real-life scenarios, such as the chances of it raining or finding a particular color of marble in a bag, can make abstract concepts more tangible. When working with numerical or geometric problems, it helps to highlight the defined outcomes with distinct labels or colors. This keeps the focus on what outcomes are beneficial to the scenario being analyzed, thus enhancing clarity and learning.

Total Number of Outcomes

The total number of outcomes is the sum of all possible results that can occur from a probabilistic event. Whether flipping a coin, rolling a die, or drawing a card from a deck, being aware of all potential outcomes is essential for calculating probabilities. This total forms the denominator of the probability fraction and is crucial for determining the relative likelihood of the desired outcome.

To educate students on identifying total outcomes, stress the importance of listing or visualizing all possibilities. For complex events, it might be necessary to create a systematic approach or use tree diagrams to ensure all outcomes are covered. Additionally, hands-on experiences, such as physically drawing from a bag of marbles, can leave a lasting impression about the range of possibilities within a given set. By emphasizing this comprehensive perspective, students will gain a stronger foundation for tackling various probability problems and scenarios.