Question

MARBLES In Exercises \(61-63\), consider a bag containing 12 marbles that are either red or blue. A marble is drawn at random. There are three times as many red marbles as there are blue marbles in the bag. What is the probability of drawing a red marble?

Short answer

The probability of drawing a red marble from the bag is \(\frac{3}{4}\).

Step-by-step solution

Determine Total Number of Marbles

Considering that the bag contains 12 marbles in total. These marbles are either red or blue.

Determine Number of Red and Blue Marbles

Based on the information given, the number of red marbles is three times the number of blue marbles. If we designate the number of blue marbles as \(x\), then the number of red marbles is \(3x\). The total number of marbles (12) is equal to the sum of red and blue marbles. So, we can write the equation as \(3x + x = 12\). Solving this equation, we find that \(x=3\). Therefore, there are three blue marbles and nine red marbles in the bag.

Calculate the Probability

To calculate the probability of drawing a red marble, we need to divide the number of favourable outcomes (number of red marbles) by the total number of outcomes (total number of marbles). So, the probability equals to \(\frac{9}{12}\), which simplifies to \(\frac{3}{4}\). That is the desired probability.

Key concepts

Understanding Probability

Probability is a measure of the likelihood that a particular event will occur. When we talk about the probability of drawing a red marble from a bag, we are really asking 'What are the chances that a red marble will be chosen over a blue one, if we pick a marble without looking?'

To determine this, we count how many favorable outcomes there are (in this case, the number of red marbles) and divide it by the total number of possible outcomes (in this case, the total number of marbles). Probability can range from 0 (the event will definitely not occur) to 1 (the event will definitely occur). In our exercise, the final probability simplified to \(\frac{3}{4}\) or 75%, meaning there is a 75% chance of drawing a red marble from the bag.

Ratios in Probability

When analyzing probability, ratios play a crucial role as they compare the number of ways a certain event can happen ('favorable outcomes') with the total number of outcomes. For example, if there are three times as many red marbles as blue marbles in the bag, this situation can be expressed as a ratio of red to blue marbles, which would be 3:1.

This ratio means that for every blue marble, there are three red ones, allowing us to set up and solve basic algebraic equations to find the exact number of each color. Understanding these ratios makes the concept of probability more intuitive and helps to quickly evaluate the chances of different outcomes.

Solving Basic Algebraic Equations

The use of basic algebra plays a significant role in solving probability problems. As seen in the exercise, to determine the number of red and blue marbles, we set up the equation \(3x + x = 12\), with \(x\) representing the number of blue marbles.

This is a simple algebraic equation that can be solved by combining like terms and isolating the variable to find its value. Once the value of \(x\) is known, it can help calculate the probability of an event occurring. It's important to become comfortable with solving these kinds of equations, as they are a fundamental part of working with probabilities in a variety of scenarios.