Imagine being so in love with mathematics that you never find yourself bored. It might sound crazy to some, but to a true math enthusiast, the possibilities are endless. **When the numbers start to blur together and your mind begins to wander, there are always fun games to play.** Take, for instance, coin flipping.

A mathematician might take a quarter from their pocket and flip it until it lands heads. But that’s not the end of the game. **They might try to figure out how long it takes to reach a certain number of flips**, or even give themselves points for each hand they play. It’s all in the name of keeping their mind sharp and their love of numbers alive.

Let’s delve deeper into the game. Suppose you flip the coin and it lands on tails (T), granting you a point. According to the rules, you continue to flip the coin until it shows heads (H). So, if you want to earn 10 points, you need to get tails in your first nine throws and heads in your last throw, resulting in a sequence of TTTTTTTTTH. If your aim is to gain just 3 points, you need to land on tails twice and heads once, in that order, giving the sequence TTH. From here on, **I’ll be using T to denote tails and H to denote heads** to simplify our foray into this mathematical philosophy.

Of course, let’s just clarify that **we’re assuming the coin we’re using is fair** — that is, the probability of landing on heads or tails is equal, 50% for each. This assumption, however, is based on the information available to us, not on a proven fact. We’ve been told that the coin is fair, but we haven’t actually tested this assertion. So, **how can we confidently assert that the coin we’re flipping is indeed fair?** **How can we verify the fairness of our coin?** This calls for a deeper understanding and exploration of probability and statistics, which I will dive into in next paragraphs.

At a first glance, it might be challenging for humans, given our inherent limitations, to discern whether a coin is fair or biased. A coin could be lumpy, with an uneven distribution of weight, which can subtly influence the outcomes of a flip. For instance, when a coin is spun, it…