Problem 13 - A Graphical Approach to College Algebra

Question

Work each problem. A marble is drawn at random from a box containing 3 yellow, 4 white, and 8 blue marbles. Find the probabilities in parts (a)-(c). (a) A yellow marble is drawn. (b) A black marble is drawn. (c) The marble is yellow or white. (d) What are the odds in favor of drawing a yellow marble? (e) What are the odds against drawing a blue marble?

Step-by-Step Solution

Answer

(a) Probability: \( \frac{1}{5} \), (b) Probability: 0, (c) Probability: \( \frac{7}{15} \), (d) Odds: \( \frac{1}{4} \), (e) Odds: \( \frac{7}{8} \).

Step 1: Calculate Total Number of Marbles

First, we need to calculate the total number of marbles in the box. To do this, we add up all the marbles: 3 yellow + 4 white + 8 blue = 15 marbles.

Step 2: Find the Probability of Drawing a Yellow Marble (a)

The probability of drawing a yellow marble is the number of yellow marbles divided by the total number of marbles. \[ P( ext{Yellow}) = \frac{3}{15} = \frac{1}{5} \]

Step 3: Find the Probability of Drawing a Black Marble (b)

There are no black marbles in the box, so the probability is zero.\[ P( ext{Black}) = \frac{0}{15} = 0 \]

Step 4: Find the Probability of Drawing a Yellow or White Marble (c)

The probability of drawing a yellow or white marble is the sum of the probabilities of drawing a yellow marble and drawing a white marble. There are 3 yellow and 4 white marbles, so: \[ P( ext{Yellow or White}) = \frac{3 + 4}{15} = \frac{7}{15} \]

Step 5: Determine the Odds in Favor of Drawing a Yellow Marble (d)

The odds in favor of drawing a yellow marble are the ratio of the number of yellow marbles to the number of non-yellow marbles. Here: \[ \text{Odds in Favor} = \frac{3}{4 + 8} = \frac{3}{12} = \frac{1}{4} \]

Step 6: Determine the Odds Against Drawing a Blue Marble (e)

The odds against drawing a blue marble are the ratio of the number of non-blue marbles to the number of blue marbles. This gives us:\[ \text{Odds Against} = \frac{3 + 4}{8} = \frac{7}{8} \]

Key Concepts

Odds calculation

Understanding odds can really help when dealing with probability problems. Odds are slightly different from probabilities. While probability is concerned with the likelihood of an event occurring, odds are a comparison of two numbers representing the occurrence and non-occurrence of an event.
Odds are expressed as a ratio. For instance, when we talk about the odds in favor of an event, like drawing a yellow marble, we compare the number of favorable outcomes (yellow marbles) with the number of unfavorable outcomes (non-yellow marbles). In the given exercise, there are 3 yellow marbles and 12 non-yellow marbles (4 white + 8 blue). Hence, the odds in favor of drawing a yellow marble are calculated as:

• Favorable outcomes: 3
• Unfavorable outcomes: 12 (the total number minus yellow marbles)

The formula becomes: \( \text{Odds in Favor} = \frac{3}{12} \), simplifying to \( \frac{1}{4} \).
Similarly, if you want to find the odds against an event, you instead compare the number of unfavorable outcomes with the number of favorable outcomes. So, the odds against drawing a blue marble take the non-blue marbles over the blue ones, yielding: \( \frac{7}{8} \). By understanding and using these simple comparisons, tackling more complex probability issues becomes much easier.

Random selection

Random selection is a fundamental concept in the world of probability and statistics. When we say random, it means each item or event has an equal chance of being selected.
In scenarios like the marble exercise, random selection means any marble in the box has an equal opportunity to be chosen. This is crucial because it ensures that the probabilities we calculate are accurate and fair.
Imagine if the process wasn't random; maybe someone always picked a marble from the top! This would skew probabilities, making some marbles more likely to be picked than others.
In our exercise, each marble color—be it yellow, white, or blue—is equally likely to be selected in a single draw, as they all reside in the box with no bias.

• Selection process: Each marble has an equal chance of being chosen.
• Equal opportunity: This fairness is what keeps probability calculations valid.
• Accuracy: Randomness ensures the calculated probabilities match reality.

Remember, ensuring randomness is vital whenever you're working on probability-based mathematics.

Marble probability

When dealing with marble probability, one must comprehend how to determine the probability of various outcomes.
Probability is the branch of mathematics that deals with calculating the likelihood of a given event's occurrence.
Probability is calculated by dividing the desired outcomes by the total possible outcomes.

In the marble exercise, knowing the total number of marbles is a vital first step. With 15 marbles, we easily calculate:

• The probability of drawing a yellow marble: \( P(\text{Yellow}) = \frac{3}{15} \)
• For a black marble, since none exist, \( P(\text{Black}) = 0 \)
• And for either yellow or white: \( P(\text{Yellow or White}) = \frac{7}{15} \) (since 3 + 4 marbles fit this category)

Knowing how to assess probability this way gives clarity. Each calculated probability reflects how likely it is for a specific event to occur based on current conditions. Practicing with marbles is a fantastic way to solidify your understanding of probability basics.