Problem 34 - Algebra for College Students

Question

Solve each problem.A jar contains seven white, six blue, and ten red marbles. If one marble is drawn at random from the jar, find the probability that (a) the marble is white or blue; (b) the marble is white or red; (c) the marble is blue or red. (a) \(\frac{13}{23}\) (b) \(\frac{17}{23}\) (c) \(\frac{16}{23}\)

Step-by-Step Solution

Answer

(a) \( \frac{13}{23} \), (b) \( \frac{17}{23} \), (c) \( \frac{16}{23} \).

Step 1: Understand the Total Number of Marbles

First, add up all the marbles in the jar. There are seven white marbles, six blue marbles, and ten red marbles. Total number of marbles = 7 + 6 + 10 = 23 marbles.

Step 2: Probability Formula

The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes. Understanding this principle will help us find the probability that a marble of a specific color is drawn.

Step 3: Calculate Probability for White or Blue Marble (a)

To find the probability that the marble is white or blue, add the number of white marbles and blue marbles. Number of favorable outcomes (white or blue) = 7 (white) + 6 (blue) = 13 marbles.Probability = \( \frac{13}{23} \).

Step 4: Calculate Probability for White or Red Marble (b)

For the probability that the marble is white or red, add the number of white marbles and red marbles.Number of favorable outcomes (white or red) = 7 (white) + 10 (red) = 17 marbles.Probability = \( \frac{17}{23} \).

Step 5: Calculate Probability for Blue or Red Marble (c)

For the probability that the marble is blue or red, add the number of blue marbles and red marbles.Number of favorable outcomes (blue or red) = 6 (blue) + 10 (red) = 16 marbles.Probability = \( \frac{16}{23} \).

Key Concepts

Events

In probability, an **event** is simply the outcome that we are interested in when a random action is performed. When you draw a marble from a jar without looking, every possible outcome, such as drawing a marble of a particular color, is considered an event.
For example, when we draw one marble from the jar, three possible events can happen:

• Drawing a white marble.
• Drawing a blue marble.
• Drawing a red marble.

Each of these events is a specific scenario that you might be interested in finding the probability for. Understanding the concept of an event is crucial as it determines what you are calculating the probability for.

Favorable Outcomes

Favorable outcomes are the specific options that align with the event you’re interested in. In the context of our exercise, these are the marbles that meet the conditions we set. When determining the probability of an event, identifying favorable outcomes is key.
Let's take a closer look at how you identify them in a practical example:

• For pulling a white or blue marble, the favorable outcomes are all white and blue marbles in the jar.
• Similarly, for a white or red event, favorable outcomes include every white or red marble in the jar.'
• And for a blue or red event, the favorable outcomes involve all blue and red marbles.

The principle is simple: count only those marbles that match the criteria of your event.

Probability Formula

The probability of an event happening is a way of measuring how likely it is for that event to take place. This is an essential part of probability theory that helps in making predictions.
The formula for calculating the probability of an event is:
\[ P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]
To apply this formula in our exercise:

• Find the total number of marbles (possible outcomes).
• Count the marbles that are favorable for your event.
• Plug these numbers into the formula to get the probability result.

By using this formula, you can solve for the likelihood of drawing either a specific combination of marble colors or any single color given the jar's composition.

Random Selection

Random selection ensures that each item (e.g., marble) has an equal chance of being chosen. This concept underpins the fairness and unpredictability of a random process.
When one marble is drawn randomly from the jar, it means that you have no knowledge or control over which marble you will pick. This random nature affects the calculation of probability since it relies on the likelihood of each individual outcome occurring.
This is why you consider all possible outcomes when calculating probability. By treating the selection as randomly fair, you can apply probability formulas correctly to predict the chance of different events happening, like "what's the probability of drawing a white marble?" Understanding the principle of random selection helps clarify why probability works the way it does in practice.