# Understanding Coin Toss Probabilities: A Mathematical Exploration

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## Chapter 1: Introduction to Coin Toss Probability

A fair coin has an equal chance of landing on heads or tails, specifically a 50% probability for each outcome. Conversely, an unfair coin will not adhere to this balance. Today, we will tackle a problem that requires determining how many times an unfair coin has been tossed based on the given data.

Before you continue, I encourage you to pause and try solving it yourself with pen and paper. Once you're ready, let's explore the solution!

### Section 1.1: Analyzing the Problem

To begin, let's assume there are n coins, out of which 2 result in heads. This indicates that n - 2 coins do not show heads. The selection of 2 coins from n is mathematically represented as:

This expression is commonly referred to as "n choose 2." To calculate the probability that 2 coins will display heads while the remaining n - 2 do not, we derive the following:

From this, we find that the probability is 3/4, derived from 1 - 1/4. Multiplying the aforementioned terms will yield the overall probability of getting 2 heads from n coins.

As a little challenge for you, can you use similar logic to determine the probability of obtaining 3 heads from n coins? Give it a shot!

### Section 1.2: Simplifying the Expressions

Did you arrive at the correct solution? Ultimately, we aim to simplify the two expressions and equate them. The formulas are as follows:

- nC2 = n(n-1)/2
- nC3 = n(n-1)(n-2)/6

Mathematically, we can express this as:

Thus, our final answer is 11!

## Chapter 2: Rewarding Your Efforts

To celebrate your hard work, here's a little treat for you:

The first video, *How Random is a Coin Toss? - Numberphile - YouTube*, delves deeper into the randomness behind coin tosses.

Next, we have another enlightening video that breaks down practical examples of coin toss probabilities:

*Probability: Toss a Coin 3 Times Example - YouTube* provides a clear illustration of how to apply these concepts.

What an intriguing journey into probability! I'm eager to hear your thoughts on this process, so please share them in the comments below.

Lastly, if you're interested in more mathematical challenges, check out the list of the best math puzzles available on Medium!

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Happy solving!