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Learn how to simulate a biased coin using Python by modifying your existing coin flip program. Understand the methods to adjust the probabilities and accurately calculate results.
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Simulating a Biased Coin in Python

Have you ever wondered how to simulate flipping a biased coin? Perhaps you are working on a project or a simulation-based course, and you want to deepen your understanding of probability. The task at hand involves simulating three coin flips and counting the number of heads. Initially, the challenge is straightforward: how do we alter the program to simulate a coin with bias, specifically a coin with a probability of heads (P(heads) = 3/4)? In this post, we will break down the solution and walk you through the steps needed to achieve your goals.

Understanding the Original Problem

The original program simulates flipping three fair coins using Python, counts the number of heads, and estimates the probability of getting 1 head. Here's how the process works:

Simulate three flips using a fair coin.

Count the number of heads formed (denote as X).

Estimate P(X = 1) and calculate E[X].

Adjust the program to allow for a biased coin where P(heads) = 3/4.

The original code leverages a random function to simulate coin flips where both heads and tails have equal probabilities of 0.5. However, to reflect a bias, we'll explore how to effectively alter this computation.

Modifying the Simulation

To effectively simulate a biased coin, you can apply one of the following strategies:

Method 1: Using random.choice with a Biased Selection

By maintaining the existing random.choice method, you can tweak the selection process. Instead of using an equal distribution, you can adjust the coin outcomes to favor heads significantly. Here's how you can modify the code:

[[See Video to Reveal this Text or Code Snippet]]

Method 2: Generating Random Floats

Alternatively, using a random float approach permits you to directly manage the probability. If you generate a random float in the range [0, 1] and check against 0.75, you can determine the outcome as follows:

[[See Video to Reveal this Text or Code Snippet]]

Method 3: Using a Probability Dictionary

For a more structured approach, you could define probabilities in a dictionary. This method provides flexibility for further adjustments and configurations. Here’s a sample structure:

[[See Video to Reveal this Text or Code Snippet]]

Final Steps

After incorporating one of the methods above into your code, you should run simulations multiple times to verify that your results align with theoretical expectations of biased coin flips. Look for outcomes where heads appear approximately 75% of the time when you count them over a large number of trials.

Ensure to incorporate the updated get_coin_surface function into the loop for counting heads in your simulations, just as you did originally.

Conclusion

Simulating biased coins can be both an enlightening and exciting exercise in probability and programming. Whether you choose to modify your existing program using selection methods, random floats, or a probability dictionary, these concepts will enhance your understanding of simulations in Python. Take your simulation further and experiment with different biases to deepen your knowledge! The world of probability awaits!