Malicious Maths and Problematic Probability

Statistics is employed in a wide range of fields. Banks, investors and gamblers exploit it to earn millions and billions of dollars every year. Everyone agrees that the right use of statistics is advantageous. However, Mr. Statistics can be tricky. I want to demonstrate this using probabilities and to warn people not to take statistics, especially probabilities, at its face value.

Let's say I have this game of dice. There's a host and a player. The game is played with the player choosing a number from 1 to 6, then tossing an unbiased 6-sided die. If the die shows a face value matching the number chosen by the player, the host pays the player $9. If it doesn't match, then the player pays the host $1. The question is, in the long run and if you are the player, is it profitable to play this game? Now many people would think, 'Hey! Why not?? An unbiased 6-sided die means each face shows 1/6 of the time! That means I will lose 5 times and win once on average, earning me $5 approximately every 6 rounds ($10 - 5x$1).' Many people would be wrong. They have forgotten that a number has to be chosen to match. Now, the probability of choosing 1 of the 6 numbers will already be 1/6. The probability of the die face showing the chosen number will be another 1/6. Combined together, the probability of getting a die face value matching your chosen number will be 1/36!!!!! (1/6 x 1/6) That means you lose $35 for every $10 win!!!

Another example: faced with an unbiased 3 choices dilemma with each choice carrying 1/3 chance to the correct answer, choosing any option gives you 1/3 of being correct while the rest of 2/3 are held by the other 2 choices. These 2 choices carry 1/3 each to the correct answer. Therefore the chance of getting the correct answer from choosing any of the other 2 is 2/3 x 1/3 = 2/9, versus 1/3 from your initial first choice. It is now statistically proven that unless there is a change, it is always the best to stay with your first choice. Similarly, imagine a scenario where the chance of getting what you want is 50%, and you only have one try. Now, you might think that getting what you want is 50% for that try. That might sounds true, but it doesn't matter. You see, let's take a large number. Say, infinite! Now, if you have infinite try, you will get what you want 50% of the time. But what matters is the one try you have now. You should then be aware that the probability of that one try carrying what you want is actually 1/4, i.e, 1/2 the probability of that try carrying what you want multiplied by 1/2 the probability of that try being one of the 50% of infinite tries that carry what you want.

So you can see, a lot of times, it's not the fault of the world that statistics lie to you. It's your fault for being ignorant of more variables. After all, stochasticity in nature is part of the wonder of this world!!! Before I end this, I would like to remind readers (or maybe reader or even no one) that I am not an expert on statistics nor do I agree on everything I have written. Also, I would like to wish a happy belated birthday to some of the special people in my life, Adeline Sung and Hui Yi, because I missed out on greeting them on time:

HAPPY BELATED BIRTHDAY!!!!

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