Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles such that at least two of them are the same color?

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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**Problem Statement:**

Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles such that at least two of them are the same color?

**Explanation:**

The problem involves calculating probabilities when selecting marbles from a jar. We're asked to find the probability that, out of three randomly selected marbles, at least two are of the same color.

**Steps to Solve:**

1. **Understand Total Possibilities:**
   - The total marbles = 6 (blue) + 3 (green) + 2 (red) = 11 marbles.
   - The total ways to choose 3 marbles out of 11 is calculated using combinations: \( C(11, 3) \).

2. **Calculate Probability:**
   - Determine scenarios where at least two marbles are of the same color:
     - Two or three marbles can be blue.
     - Two or three marbles can be green.
     - Two or three marbles can be red.
   - These scenarios will include:
     - Two blues and one of another color.
     - Two greens and one of another color.
     - Two reds and one of another color.
     - Three blues, three greens, or three reds.

3. **Use Complementary Probability:**
   - Alternatively, compute the probability of selecting three marbles all of different colors and subtract from 1.
   - Calculate using combinations for the number of ways to choose one marble of each color.

4. **Result Interpretation:**
   - The result will give the probability of picking at least two marbles with the same color from the jar.

This step-by-step approach helps in understanding how probability is applied in situations with multiple conditions.
Transcribed Image Text:**Problem Statement:** Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles such that at least two of them are the same color? **Explanation:** The problem involves calculating probabilities when selecting marbles from a jar. We're asked to find the probability that, out of three randomly selected marbles, at least two are of the same color. **Steps to Solve:** 1. **Understand Total Possibilities:** - The total marbles = 6 (blue) + 3 (green) + 2 (red) = 11 marbles. - The total ways to choose 3 marbles out of 11 is calculated using combinations: \( C(11, 3) \). 2. **Calculate Probability:** - Determine scenarios where at least two marbles are of the same color: - Two or three marbles can be blue. - Two or three marbles can be green. - Two or three marbles can be red. - These scenarios will include: - Two blues and one of another color. - Two greens and one of another color. - Two reds and one of another color. - Three blues, three greens, or three reds. 3. **Use Complementary Probability:** - Alternatively, compute the probability of selecting three marbles all of different colors and subtract from 1. - Calculate using combinations for the number of ways to choose one marble of each color. 4. **Result Interpretation:** - The result will give the probability of picking at least two marbles with the same color from the jar. This step-by-step approach helps in understanding how probability is applied in situations with multiple conditions.
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