Problem 47 - Algebra for College Students

Question

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:One red and one white marble are drawn. \(\frac{15}{34}\)

Step-by-Step Solution

Answer

The probability is \( \frac{15}{34} \).

Step 1: Identify Total Marbles and Initial Probability

The bag contains a total of 17 marbles (5 red and 12 white). The probability of initially drawing a red marble is \( \frac{5}{17} \) since there are 5 red marbles out of 17.

Step 2: Calculate Probability of Red Then White

If a red marble is drawn first, 16 marbles are left with 12 of them being white. The probability of then drawing a white marble is \( \frac{12}{16} \) or \( \frac{3}{4} \). Thus, the combined probability of drawing a red marble first and then a white marble is \( \frac{5}{17} \times \frac{3}{4} = \frac{15}{68} \).

Step 3: Calculate Probability of White Then Red

Alternatively, if a white marble is drawn first, there are now 16 marbles remaining with 5 being red. The probability of drawing a red marble next is \( \frac{5}{16} \). Therefore, the combined probability of drawing a white marble first and then a red marble is \( \frac{12}{17} \times \frac{5}{16} = \frac{15}{68} \).

Step 4: Add Probabilities of Both Scenarios

To find the probability of drawing one red and one white marble in any order, add the probabilities from both scenarios: \( \frac{15}{68} + \frac{15}{68} = \frac{30}{68} = \frac{15}{34} \).

Step 5: Verification: Confirm Simplification

Ensure that \( \frac{30}{68} \) simplifies correctly to \( \frac{15}{34} \). Since both numerator and denominator are divisible by 2, the simplification is correct.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and understanding patterns. In probability, it helps us manage and predict combinations and outcomes. For example, if you have a set of different colored marbles, combinatorics allows you to calculate possible arrangements and selections.
In probability exercises like the marble draw, combinatorics is useful to calculate different ways you can draw marbles. This involves understanding permutations (order matters) and combinations (order does not matter). Here, permutations are relevant because the order of drawing affects the outcome.

• There are 17 marbles, giving us a total set size.
• We proceed with permutations because drawing marbles affects the order of probability calculation.

By breaking down potential sequences of events, we can accurately calculate overall probabilities.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already taken place. It's expressed as \( P(A|B) \), the probability of event A happening given that B has happened. This is crucial in scenarios where events are dependent on each other.
In the marble problem, after a marble is drawn, the probability calculations rely on this concept. Drawing one marble is an event that alters the chances for the subsequent draw. For instance:

• If a red marble is drawn, the probability of then drawing a white is \( \frac{12}{16} \) since there are 16 marbles left.
• Conversely, if a white marble is taken first, the probability of picking a red next becomes \( \frac{5}{16} \).

This illustrates how initial events modify probabilities for subsequent events.

Mathematics Education

Teaching probability theory effectively involves simplifying complex ideas into digestible parts. Engaging students with real-world examples, like a marble drawing exercise, makes abstract concepts more relatable and understandable.
We promote understanding by encouraging students to:

• Break down problems into steps, making them less intimidating.
• Practice simplifying fractions and apply real-life scenarios to theoretical concepts.
• Explore different methods of calculation, such as tree diagrams, to visualize outcomes.

Ultimately, fostering these skills in mathematics education encourages logical reasoning and enhances problem-solving abilities, making learning rewarding and accessible.