Problem 44 - Algebra for College Students

Question

A bag contains one red and two white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are red. 0

Step-by-Step Solution

Answer

The probability of drawing two red marbles is 0.

Step 1: Identify the Total Outcomes

To determine the total number of outcomes when drawing two marbles one after the other without replacement, first, note that the bag contains 1 red and 2 white marbles, making a total of 3 marbles. The possible ways to draw two marbles out of these 3 marbles can be calculated using combinations:\[\text{Total outcomes} = \binom{3}{2} = 3\]

Step 2: Identify Favorable Outcomes

For both marbles drawn to be red, we would need to draw two red marbles. However, since there is only 1 red marble in the bag, it is impossible to draw 2 red marbles out of it. Thus, there are 0 favorable outcomes for this event.

Step 3: Calculate the Probability

The probability of an event is given by the ratio of favorable outcomes to total possible outcomes. In this case, as calculated:\[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{0}{3} = 0\]

Key Concepts

Understanding Combinations

In probability, combinations help us determine how many unique sets we can create from a larger group. When dealing with combinations, we are interested in the possible ways of selecting items where the order does not matter. For example, in our marble problem, we have a bag with 3 marbles, and we're drawing 2 marbles without replacement.
To figure out how many ways we can select these two marbles, we use a mathematical formula for combinations:

• Combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. In our case, \( n = 3 \) and \( r = 2 \). The solution becomes:\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]This tells us there are 3 unique ways to draw 2 marbles from the bag of 3 marbles.

Exploring Favorable Outcomes

Favorable outcomes in probability refer to the specific results that we are interested in when assessing the likelihood of an event. It’s essential to identify how many of these desired outcomes exist within the total possibilities.
In the context of our marble problem, we want to find how likely it is to draw two red marbles. However, there is only 1 red marble in the bag. Because it is not possible to draw 2 reds from a single red marble, the number of favorable outcomes is 0.
Recognizing favorable outcomes clarifies the probability calculation process and allows us to understand why some events have specific probabilities. In this case, because there are no favorable outcomes, the likelihood of pulling two red marbles is zero. This knowledge helps in understanding limitations in scenarios with limited resources.

Analyzing the Marbles Probability Problem

Probability problems, like our marbles example, often involve determining potential outcomes and understanding likelihoods based on constraints, such as replacement or lack thereof.
The main formula for probability is:

• Probability formula: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)

Here, we previously determined that the total number of outcomes was 3 and the number of favorable outcomes was 0. Substituting these values into the formula gives: \[ P = \frac{0}{3} = 0 \]This solution demonstrates the probability of drawing two red marbles is zero since it wasn't feasible given the marble distribution.
Problems like these teach us valuable lessons about determinability and constraints, illustrating that sometimes certain desired results, like getting two red marbles, aren’t possible. Understanding these fundamental probability concepts can help you tackle more complex problems in real-world scenarios.