# Unraveling a Probability Puzzle with Bayes’ Theorem

Written on

## Chapter 1: The Probability Challenge

Imagine you're presented with a treasure chest from which you randomly select a coin. If the coin turns out to be gold, what is the likelihood that you originally chose chest A? While this problem might not seem overly complex, it serves as a great example for applying Bayes’ Theorem.

Before moving forward, take a moment to grab some paper and a pen. Try to solve this challenge on your own before checking the solution!

### Solution 1: An Intuitive Approach

To understand this intuitively, consider the two possible sources for the gold coin: chest A, which guarantees a gold coin, and chest B, where there is only a 50% chance. Given that we are twice as likely to pick a gold coin from chest A, the odds favor it at a ratio of 2:1. Thus, the probability of having selected chest A is calculated to be 2/3.

### Solution 2: Applying Bayes’ Theorem

Now let’s analyze this scenario using Bayes’ Theorem. The question we need to answer is: what is the probability of having chosen chest A, given that we picked a gold coin?

The mathematical representation of this is:

This expression essentially seeks the probability of event A occurring, given that event G has already taken place—specifically, the event of choosing chest A after obtaining a gold coin.

Utilizing Bayes’ Theorem, the formula can be expressed as:

Here, we can break it down into:

- P(G|A) = 1, since all coins in chest A are gold.
- P(G) = 150/200, as there are 150 gold coins among 200 total coins.
- P(A) = 1/2, since there are two chests to choose from.

Plugging these values into the equation gives us:

And thus, we arrive at the answer!

### Bonus Challenge: A Twist in the Tale

Here's an intriguing bonus problem: A family has two children. If one of them is a boy, what is the probability that both children are boys? Share your reasoning in the comments. If you solve it correctly, you might just uncover the next treasure chest!

What was your thought process this time around? I’m eager to hear your insights in the comments!

In the video titled "Could you solve this hard probability question?" you’ll find further explorations of probability challenges similar to this one, perfect for sharpening your skills.

## Chapter 2: Diving Deeper into Probability

In "Probability Word Problems (Simplifying Math)", you can learn how to simplify and tackle word problems involving probability, enhancing your understanding of the concepts.

## Math Puzzles Collection

If you're interested in more challenging math puzzles, check out this curated list on Medium, featuring topics across Algebra, Geometry, Calculus, and Number Theory. Share these with your friends and enjoy solving together!

Thank you for reading! If you found this article insightful, please give it a clap. Your support encourages me to continue writing engaging content.

If you appreciate the effort I put into these articles, consider buying me a coffee. Your generosity supports my writing journey and academic pursuits.

Happy Solving!