Question

On a nearby pond, black and white ducks are swimming in groups of three. James wants to find the experimental probability of two white ducks and one black duck swimming together. Design a simulation using a coin flip and explain why it is the best choice for James.

Answer 1

To simulate the situation of ducks on a pond using a coin flip, we’ll create a system where each flip of the coin represents the chance of observing either a black duck (B) or a white duck (W). The rationale for using a coin is that it provides a simple binary outcome, just like the two types of ducks. Here’s how you can design the simulation:

### Simulation Design

1. **Assign Values to Coin Sides:**
- Assign "Heads" (H) to represent a white duck (W).
- Assign "Tails" (T) to represent a black duck (B).

2. **Simulating Groups of Ducks:**
- For each group of three ducks, perform three coin flips. Each flip corresponds to one duck.
- Record the outcome of each flip (either H for a white duck or T for a black duck).

3. **Repeat the Simulation:**
- Conduct the simulation for a large number of trials (e.g., 1000 trials).
- Count how many times you observe two white ducks and one black duck (the combination you are interested in).

### Example of a Single Trial
Suppose you flip the coin three times and get the following results:
- Flip 1: H (W)
- Flip 2: H (W)
- Flip 3: T (B)

This would be recorded as a successful outcome because it has two white ducks and one black duck.

### Calculating Experimental Probability
After conducting all trials:
- Let \( N \) be the total number of trials (e.g., 1000).
- Let \( S \) be the number of successful outcomes that match the combination of two white ducks and one black duck.
- The experimental probability \( P \) can be calculated using the formula:

\[
P(\text{2 W, 1 B}) = \frac{S}{N}
\]

### Why This Simulation is the Best Choice for James
1. **Simplicity:** A coin flip is straightforward to set up and execute. It only requires a coin and can be easily repeated.
2. **Binary Outcomes:** Each coin flip represents a clear dichotomy (white or black duck), mirroring the problem of having two types of ducks.
3. **Random Sampling:** The randomness of flipping a coin effectively simulates the random arrangements of ducks swimming together.
4. **Scalability:** You can perform as many trials as desired to increase the accuracy of the estimated probability.
5. **Statistical Relevance:** Using a probabilistic model, it can yield results that converge to the theoretical probability when enough trials are conducted, thereby giving James a reliable understanding of the expected outcomes.

Overall, this method provides a practical and effective way for James to estimate the probability of seeing two white ducks and one black duck in groups of three.