2 coin

Guri Bee

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22-10-2024

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The Two-Coin Problem: A Dive into Probability with a Simple Setup

The "Two-Coin Problem" is a classic probability puzzle that, despite its apparent simplicity, can lead to surprising results. It's a great way to understand the fundamental concepts of probability and how seemingly straightforward scenarios can yield unexpected outcomes.

The Problem:

You have two coins. One is a fair coin (heads and tails with equal probability), and the other is a biased coin with heads appearing 75% of the time. You randomly select one of the coins and flip it. The result is heads. What is the probability that the coin you selected is the biased coin?

Understanding the Solution:

This problem can be tackled using Bayes' Theorem, a fundamental tool in probability. Here's a breakdown of how it works:

1. Define Events:

   • A: The event of selecting the biased coin.
   • B: The event of getting heads.

2. Prior Probabilities:

   • P(A): The probability of selecting the biased coin, which is 1/2 (50%) as you have an equal chance of picking either coin.
   • P(¬A): The probability of selecting the fair coin, which is also 1/2 (50%).

3. Conditional Probabilities:

   • P(B|A): The probability of getting heads given you selected the biased coin, which is 0.75 (75%).
   • P(B|¬A): The probability of getting heads given you selected the fair coin, which is 0.5 (50%).

4. Bayes' Theorem:

   Bayes' Theorem states: P(A|B) = [P(B|A) * P(A)] / P(B)

   We need to find P(B), which is the probability of getting heads regardless of the coin selected. We can use the law of total probability:

   P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A) P(B) = (0.75 * 0.5) + (0.5 * 0.5) = 0.625

5. Calculating the Probability:

   Now we can plug the values into Bayes' Theorem:

   P(A|B) = (0.75 * 0.5) / 0.625 = 0.6

Conclusion:

The probability that you selected the biased coin given that the flip resulted in heads is 0.6 or 60%. This is higher than the initial 50% probability of selecting the biased coin. This illustrates how observing an event (getting heads) can influence our belief about the underlying cause (which coin was chosen).

Additional Considerations:

This problem can be extended with more coins or different bias percentages. The key takeaway is that Bayes' Theorem provides a framework for updating our beliefs based on new evidence.

Practical Applications:

The Two-Coin Problem, though seemingly simple, has real-world applications in fields such as:

• Medical Diagnosis: Determining the probability of a disease given a specific symptom.
• Spam Filtering: Identifying emails as spam based on certain characteristics.
• Machine Learning: Training models to predict outcomes based on observed data.

References:

• This example and explanation are inspired by various sources, including online forums and textbooks on probability.
• Credit for the problem setup goes to numerous discussions on platforms like Stack Exchange and Reddit.

Note: Please note that this explanation and analysis are for educational purposes and may not be comprehensive. For in-depth study of probability and Bayes' Theorem, refer to specialized textbooks and resources.