de Mere's paradox: how mathematics impacts the world more directly than we think it does

de Mere's paradox: how mathematics impacts the world more directly than we think it does
StuffThatMatters: Science & Mathematics 

Mathematics is everywhere – from simple, day-to-day calculations involving profits and losses to the complicated equations that reveal to us the profound nature of the depths of reality, in the form of quantum mechanics. However, little do we use our mathematical reasoning to make sense of this seemingly senseless world. As did Chevalier de Mere, of France, a mathematician who liked to bet in French salons and bars, for the sake of understanding the probabilities behind betting. 

Mere was quite fascinated with particularly one problem – the bet that we'd see a 6 come up when a dice is rolled 4 times, put against the bet that 2 sixes will come up when a pair of die are rolled 24 times. Naturally, Mere concluded that the probabilities of both these events were equal, on the basis of mistaken empirical probability, and he did regret his mistake. 

And interestingly, this problem also started modern probability theory. 

Initial solution

We all know that a die has 6 faces, with numbers 1-6 embedded on each side. So, when we throw a die, there are 6 possible outcomes. The probability of a 6 coming up would be: 

P ( a 6 comes up ) = number of desirable outcomes / number of possible outcomes
= 1 / 6

And similarly, when we roll 2 dice simultaneously, there are 6 x 6 possible outcomes = 36 outcomes. 

So, P ( a double-six comes up ) = 1 / 36

Now, Mere concluded that since 4 / 6 = 24 / 36, 

the probabilities of both the events are equal. However, he observed the following: 

1. When he took the first bet, i.e betting that a 6 would come up on rolling a single die for 4 times, he won most of the time, and gained money.

2. Growing confident, Mere betted that a double-six would turn up, when a pair of dice were rolled for 24 times, and he lost lots and lots of money! The more he played, the angrier he got seeing that he was losing most of the time. And this continued for a long time.

Pascal and the correct interpretation

Frustrated, de Mere contacted Pascal, who then correctly framed the problem. Let us look at it more clearly, using the same classical model of empirical probability. 

For the first scenario, we roll a single die for 4 times. Naturally, the number of possible, equiprobable (i.e equally possible) outcomes is 6 each time. Now, since the experiment is done 4 times, i.e the die is rolled 4 times, the number of equaprobable outcomes is 6 x 6 x 6 x 6 = ( 6 ) ^ 4

Now, each time, if we don't get a 6, we lose. So, the probability of getting a 6 each time is 1 / 6, but the probability of NOT GETTING a 6 is 1 – 1 / 6 = 5 / 6

When the die is rolled 4 times, the probability of not getting a 6 in the process becomes: 

( 5 / 6 ) x ( 5 / 6 ) x ( 5 / 6 ) x ( 5 / 6 ) 
= ( 5 / 6 ) ^ 4

So, the probability of winning the bet, i.e a 6 turning up in the experiment, is: 

P ( Getting a 6 ) = 1 – P ( Not getting a 6 )
= 1 - ( 5 / 6 ) ^ 4 
= 1 – 5^4 / 6^4
= 1 – 625 / 1296
= 671 / 1296 
= 0.5177 ( approximately) > 0.5 

So, P ( Getting a 6 ) > 1/2, i.e when the experiment is repeated for a large number of times, it is more probable that the bet will be won, since there's MORE THAN 50% chance that it will be won. 

The second scenario: why de Mere lost the bet 

Now, for the second scenario, for each time the pair of dice is rolled, we have 6 possible outcomes for the first die in the pair, and 6 for the second die. So, there are 6 x 6 = 36 possible combinations = 36 equiprobable outcomes. 

Now, the probability of getting a pair of sixes is 1 / 36 , and so the probability of not getting that would be: 

P ( Not getting a pair of sixes ) = 1 – 1 / 36
= 35 / 36 

And so, when the pair of dice is rolled for 24 times, we are conducting the experiment 24 times. So, the probability of not getting the desired result in 24 rolls would be: 

P ( Not getting a pair of sixes in 24 rolls ) = ( 35 / 36 ) x ( 35 / 36 ) x ... (24 times)
= ( 35 / 36 ) ^ 24
=  11419131242070580387175083160400390625 / 22452257707354557240087211123792674816
= 0.5086 (appx)

So, P ( Getting a pair of sixes in 24 rolls ) = 1 – 0.5086 
= 0.4914

Naturally, P ( Getting a pair of sixes in 24 rolls ) < 1/2, 

And so, there's less than 50% chance of winning the second bet, i.e most of the time the bet will be lost when the pair is rolled for a large number of times. 

A mathematical philosophy? 

Blaise Pascal, one of my many heroes, was also a philosopher. And his philosophy involved the use of mathematics, to make bets. Because Pascal, as we know, used to see the reality itself as a betting table. He used mathematics to justify the belief in the Christian God, famously. While that justification is rightly challenged and disputed, his use of mathematics, for solving the crises we face while taking decisions in daily life, is noteworthy. 

Pascal's philosophy involed what is known as the rational utilization of expected values, or E.Vs, to make decisions. Whenever we have to make a choice, we're virtually gambling, because our course of action is much like a bet against the unknown. And how should we place this bet? Much like de Mere's first bet, with a positive E.V (or +EV). 

That is, we should take decisions which have a greater than 50% probability of being successful. And if a decision has a -EV, i.e less than 50% chance of getting successful, then we should possibly abandon it. 

It's all a very general, very brief and very rough explanation of Pascal's philosophy, mind you, but the important thing is to realize that he managed to combine philosophy and mathematics, thus providing a more concrete reasoning behind ethical problems. 

Conclusion 

This problem, named after de Mere, is indeed a veridical paradox, as ProofWiki notes. I already used the term in my previous posts, and these are just cases in which the premise is simple to understand, but the outcome is counter-intuitive, i.e against what we'd commonly believe. 

This shows us yet another scenario of a self-evident fact, that the universe is mathematical. More philosophically speaking, the reality itself seems to prefer mathematics. Maybe the Anthropic principles will tell us why, some day. Till then, take care. 

References

http://ift.tt/1xWu9mg 

Magnificent Mistakes in Mathematics (Alfred Posamentier) 

http://ift.tt/1xWubdZ

#math   #interesting   #science   #probability  




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