# Exploring the Probability of Flipping 50 Heads in 100 Tosses

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## Understanding the Probability Challenge

Today's mathematical exploration involves flipping a fair coin 100 times. The objective is to determine the likelihood of obtaining exactly 50 heads.

Before you dive in, grab a pen and paper, and try solving it yourself! Once you’re ready, continue for the solution!

### The Problem Breakdown

Imagine arranging the outcomes of each coin flip in a row, from left to right. Each flip results in either heads or tails.

In this scenario, the probability of any specific arrangement of 100 flips is (1/2)¹⁰⁰, since the tosses are independent and fair.

For instance, the chance of the first 50 flips being heads and the subsequent 50 also being heads is equivalent to the probability of all 100 being heads!

This leads us to a fundamental question:

### Using Binomial Coefficients

Fortunately, we can leverage binomial coefficients to determine how many ways we can select 50 heads from 100 flips.

This is expressed as "100 choose 50." By multiplying this by (1/2)¹⁰⁰, we can calculate the probability of achieving exactly 50 heads.

Consequently, the odds of obtaining an equal number of heads and tails are quite slim!

### Reflections on the Solution

What do you think about this problem? I’d love to hear your thoughts in the comments below!

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## Chapter 1: Probability in Coin Tosses

This video explores the concept of microstates and the probability of flipping heads and tails when you toss 50 fair coins.

## Chapter 2: Exact Outcomes in Coin Tossing

In this video, we delve into the probabilities of achieving specific outcomes during multiple coin tosses.