The history of probability theory is introduced. We visit ideas developed by Cardano, Pascal and Fermat. Gambling motivated an evolved perspective on random sequences of events. The concept of frequency stability will play an important role years later in our story.
Transcript
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[Music] ship in seven years is almost his retired fight quite cool fighting from the perance all of these games are structured worlds with well- defined rules each involves strategies and potential payoffs when we put money on games the outcome affects us therefore we care about the odds the understanding of odds or chance leads to probability Theory the founding father of this subject had a serious stake in understanding the outcome of a game of dice cardano was an Italian mathematician with a well-known gambling addiction he wrote letters in which he bragged about his ability to beat his friends his trick was to place his bets using his ideas from mathematics his breakthrough was a method of calculating the certainty or probability of some random event such as rolling Snake Eyes let's pause for a moment and think about the physics of dice there are two reasons why a dice roll is unpredictable or random the first is symmetry dice are designed to be symmetrical unlike an egg which is asymmetrical always falling predictably onto its broad side dice are balanced so they do not favor a side the second is mixture each time we roll a dice tiny changes in the initial position and velocity of the dice are Amplified as it bounces along its path the unpredictability comes from not knowing the exact initial speed position and direction of the toss this results in a powerful property cardono noticed every outcome is equally likely this allowed him to calculate the probability by developing what's now called a probability space first he counts all the possible outcomes then he defines the event in question such as rolling a one which can occur in one way the probability is then found by dividing the event by all possible events which in this case works out to six realize the cold calculating logic here there is no such thing as a lucky number no divine intervention the probability of rolling any number is exactly 16 the same log applies when we roll multiple dice imagine he needed to know the probability of rolling a pair such as Snake Eyes First he counts the size of the sample space with two dice rolls there are 36 possible outcomes six * 6 then he counts the number of ways you can roll a pair there are six different pairs so 6 / 36 is the probability of rolling a pair also 16 this simple yet powerful idea allowed cardano to bet according to the true odds while his opponents Place their bets based on hunches and lucky numbers remember this works with multiple tosses imagine we needed to know the odds of rolling three ones simple first we figure out the size of the sample space for three dice this is 6 * 6 * 6 or 216 there is only one way to roll 3 ones so the probability is 1 / 216 this was the trick it was not based on Magic but mathematics years later two more Italians Pascal and fermat refined cardano's idea while they were pondering the outcome of sequences of random events such as multiple coin flips consider the following imagine two rooms inside each room room is a switch in one room there is a man who flips his switch according to a coin flip if he lands heads the switch is on tails the switch is off in the other room a woman switches her light based on a blind guess she tries to simulate Randomness without a coin then we start a clock and they make each switch in unison now the interesting question question can you determine which light bulb is being switched by a coin flip the answer is yes but how the first step is to keep track of each sequence of events 1 equals on 0 equals off the trick is to think about the properties of each sequence rather than looking for specific patterns first we may try to count the number of ones in zeros in each sequence this is close but not enough since they will both seem fairly even the answerers to count sequences of numbers such as runs of three consecutive switches a true random sequence will be equally likely to contain every sequence of any length this is called the frequency stability property and is demonstrated by this uniform looking graph the forgery now is obvious humans favor certain sequences when they make guesses resulting in uneven patterns such as we see here one reason this happens is because we make the mistake of thinking certain outcomes are less random than others but realize there is no such thing as a lucky number and no such thing as a lucky sequence if we flip a coin 10 times it is equally likely to come up all heads all Tails or any other sequence you can think of