At a county courthouse, 42 citizens have been summoned for potential jury duty. Three of the summoned citizens are brothers. The court randomly assigns 14 citizens each to one of three courtrooms. Determine the probability that each brother is assigned to a different courtroom. Express probability as a percentage rounded to two decimal places, ie.: 0.1885 as 18.85%.

We are given a situation in which 42 citizens, including three brothers, are randomly assigned to three courtrooms. We want to find the probability that each of the three brothers is assigned to a different courtroom.

### Step 1: Total possible assignments
The three brothers can be assigned to any of the three courtrooms, and each brother can independently be assigned to any of the rooms. Thus, for the three brothers, the total number of possible assignments (without any restrictions) is:

\[
3 \times 3 \times 3 = 27
\]

This is because each of the 3 brothers has 3 choices of courtrooms.

### Step 2: Favorable assignments
Now, we want to count the favorable outcomes, where each brother is assigned to a different courtroom. This is essentially the number of ways to assign the 3 brothers to 3 distinct courtrooms, which is equivalent to the number of ways to permute the 3 brothers among the 3 courtrooms:

\[
3! = 6
\]

### Step 3: Probability calculation
The probability that each brother is assigned to a different courtroom is the ratio of favorable outcomes to total possible outcomes:

\[
\text{Probability} = \frac{6}{27} = \frac{2}{9}
\]

### Step 4: Convert to percentage
To express this probability as a percentage, we multiply by 100:

\[
\text{Probability as a percentage} = \frac{2}{9} \times 100 \approx 22.22\%
\]

Thus, the probability that each brother is assigned to a different courtroom is **22.22%**.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How does the probability change if more siblings are involved?
2. What if the courtrooms are not equally populated?
3. Can you apply this method to other similar assignment problems?
4. How would the probability change if the number of courtrooms increases?
5. What if the brothers must be assigned to the same courtroom?

**Tip:** In probability problems, it's important to clearly define favorable outcomes and total outcomes to set up the ratio correctly.

Find the probability that three brothers summoned for jury duty are assigned to different courtrooms out of 42 total citizens. This problem involves key probability concepts, combinatorics, and the use of permutations. Ideal for students in Grades 10-12, the solution shows how to calculate the probability step by step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation