Adding Infinite Numbers: A Probabilistic Approach to Mathematical Philosophy
It’s really hard for someone who loves mathematics to get bored. That’s because mathematicians can play fun games when they get bored. For instance, they can play coin flipping with a quarter they take off from their pocket till they land heads and try to figure out how long it takes them to play the game. They can even improve the game and give themselves a point for each hand they play. Let’s suppose that you threw the coin, and you got tails. In that case, you earn a point and throw the coin repeatedly until you get heads. So, to earn 10 points, your first nine throws should be tails, and your last throw should be heads. Or, to earn 3 points, you should respectively get tail-tail-head (TTH). From now on, I will refer to tails with T and heads with H.
Of course, by the way, we accept that the coin we are using is fair, which means that the probability of getting heads or tails is equal, and 50% as agreed. However, we could only assume the coin is fair because we don’t know if the coin is fair or biased. We are only told that the coin is fair, and we haven’t proved it yet. So how can we be sure that the coin is fair? In other words, how can we prove that the coin we are betting a flip is fair?
First of all, since we are very simple creatures as humans, we can never understand whether a coin is fair or biased by just looking at it. Any coin might be slightly lumpy, with a highly non-uniform distribution of weight; in fact, a spinning coin tends to fall toward the heavier side more often. For instance, flipping an American quarter is pretty close to being fair, but it is biased toward whatever side was upside when the coin was thrown into the air. Certain mathematicians have proven that the ratio is 51/49, not 50/50, which comes in handy to some prisoners in America who know this fact and use the information to earn money by playing the coin-flipping game.
We might try to put the coin upright because it couldn’t stand upright and topple if it is biased. However, if the non-uniform distribution of weight is very low, the coin can still stand upright.
