Twenty-seven athletes are randomly divided into three heats of nine for a 100m sprint. Given that Alfred is in a different heat from Richardson, and Richardson is in a different heat fres Jefferson, what is the probability that Alfred and Jefferson are in the same heat

To solve this problem, we need to determine the probability that Alfred and Jefferson are in the same heat, given the constraints about Alfred, Richardson, and Jefferson being in different heats. Here's a step-by-step explanation of how we can calculate this probability.

### Step 1: Understand the Constraints

We have 27 athletes divided into 3 heats (A, B, and C), each containing 9 athletes. The conditions are:
1. Alfred (A) and Richardson (R) are in different heats.
2. Richardson (R) and Jefferson (J) are in different heats.

We need to find the probability that Alfred (A) and Jefferson (J) are in the same heat.

### Step 2: Distribute Richardson

We can place Richardson (R) in any of the 3 heats. Let's assume Richardson is in Heat 1. The calculations will be similar if Richardson is placed in Heat 2 or Heat 3, due to symmetry.

### Step 3: Place Alfred

Since Alfred cannot be in the same heat as Richardson, Alfred must be placed in either Heat 2 or Heat 3. There are 2 choices for placing Alfred.

### Step 4: Place Jefferson

Given that Jefferson must be in a different heat from Richardson, there are two possible placements for Jefferson:
1. Jefferson can be in the same heat as Alfred.
2. Jefferson can be in the remaining heat.

### Step 5: Calculate the Probability

Given the above setup, let's summarize the possible heat placements:

1. Richardson (R) is in Heat 1.
2. Alfred (A) has 2 choices: Heat 2 or Heat 3.
3. Jefferson (J) has 2 choices for each placement of Alfred:
   - If Alfred is in Heat 2, Jefferson can be in Heat 2 or Heat 3.
   - If Alfred is in Heat 3, Jefferson can be in Heat 3 or Heat 2.

To find the probability that Alfred and Jefferson are in the same heat:
- There are 2 cases where Alfred and Jefferson can be in the same heat:
  1. Alfred in Heat 2 and Jefferson in Heat 2.
  2. Alfred in Heat 3 and Jefferson in Heat 3.

Since Jefferson has 2 choices in each of the two possible placements of Alfred, there are a total of 4 possible placements (2 for each heat). Out of these, 2 are favorable (where Alfred and Jefferson are in the same heat).

Thus, the probability \( P \) is:
\[
P(\text{A and J in the same heat}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{2}{4} = \frac{1}{2}
\]

However, this calculation did not account for all possibilities because we considered placements symmetrically. To correct the calculation, we note that:
\[
P(\text{A and J in the same heat}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]

Therefore, the corrected probability that Alfred and Jefferson are in the same heat, given the constraints, is:
\[
\boxed{\frac{1}{4}}
\]

Would you like more details or have any questions?

Here are 8 relative questions for further exploration:
1. How does the calculation change if we have more athletes or heats?
2. What if there were additional constraints on which heats athletes could be placed in?
3. How do we calculate the probability if Alfred, Richardson, and Jefferson are randomly assigned without constraints?
4. Can we apply this method to other types of sports or competitions with different rules?
5. What is the impact of having uneven heat sizes on the probability calculation?
6. How does the probability change if there are more than three key athletes with constraints?
7. What other probability problems involve partitioning a set into subsets?
8. How can this approach be applied to scheduling problems in operations research?

**Tip:** In probability problems with constraints, breaking down the problem and considering all possible scenarios can help in systematically determining the correct probability.

Explore a probability problem involving 27 athletes randomly divided into three heats for a 100m sprint. Learn how to calculate the probability that specific athletes are in the same heat given constraints. Suitable for Grades 10-12.

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