Brilliant Mathematician's Fallacy

Jean le Rond D'Alembert ( the Alembert system is named for him - a type of martingale) was a brilliant mathematician who created a great deal of controversy by stating this in one of his major works around the the year 1750. It would be interesting to here some of the mathletes on the site comment about this, although I guess it is fairly easy to prove him wrong. From the linked article:

" He had been made responsible for the mathematical section of this enterprise, and it was in this capacity that he wrote his first contributions to the calculus of probabilities. Following some indirectly critical articles on this topic a scandal broke out about the article on heads or tails, (croix ou pile) in volume IV of the "Encyclopédie. D'Alembert contested principles accepted by everyone: he cast doubt, for example, on the fact that the probability of obtaining heads in two trials was 3/4, and suggested that it might be 2/3, arguing that there were in fact only three possible events (the coin is not thrown again if heads occurs at the first trial), namely heads, tails followed by heads, and tails followed by tails, the first two being favourable, while the third was not. D'Alembert was thus provocatively initiating a prickly debate on the question of equally likely events."

https://www.encyclopediaofmath.org/index.php/DAlembert

FWIW, if you search this site (WoV) for d'Alembert, you will find dozens of references to it and several threads with pretty comprehensive discussion.

Quote: beachbumbabs

FWIW, if you search this site (WoV) for d'Alembert, you will find dozens of references to it and several threads with pretty comprehensive discussion.

thanks but that is not what I was interested in. I don't think the d'Alembert system might be a winner.


according to the link d'Alembert:


"he cast doubt, for example, on the fact that the probability of obtaining heads in two trials was 3/4, and suggested that it might be 2/3, arguing that there were in fact only three possible events (the coin is not thrown again if heads occurs at the first trial), namely heads, tails followed by heads, and tails followed by tails, the first two being favourable, while the third was not."


the way I understand this is that the established mathematicians said the probability of obtaining one heads in 2 coin flips was 75%

and d'Alembert said no, it was 66.6% because of what is stated above


I wanted to know how to prove d'Alembert wrong in this particular example

Well, there are several ways to think about it.
First of all, just because there are a specific number of possibilities (3) does not mean that they occur with equal probability (1/3)

So there are a maximum of two coin tosses to get one head.
On the first throw, 1/2 will be a head and 1/2 will be a tail.
However, if you find that you have thrown a tail, you now need to throw again.
So, of the 50% of the throws that start with a tail, 50% will be followed by a head, and 50% will be followed by a tail.
Half of 50% is 25%

With all that, you end up with three possibilities with different probabilities:
H 50%
TH 25%
TT 25%

You can see I have accounted for all three possibilities with a total probability of 100%

If you now add up the PROBABILITIES (not possibilities) of rolling a head, 50%H plus 25%TH = 75%

Quote: beachbumbabs

FWIW, if you search this site (WoV) for d'Alembert, you will find dozens of references to it and several threads with pretty comprehensive discussion.

Also the term mathlete has been discussed as derogatory

Quote: Dalex64

Well, there are several ways to think about it.
First of all, just because there are a specific number of possibilities (3) does not mean that they occur with equal probability (1/3)

So there are a maximum of two coin tosses to get one head.
On the first throw, 1/2 will be a head and 1/2 will be a tail.
However, if you find that you have thrown a tail, you now need to throw again.
So, of the 50% of the throws that start with a tail, 50% will be followed by a head, and 50% will be followed by a tail.
Half of 50% is 25%

With all that, you end up with three possibilities with different probabilities:
H 50%
TH 25%
TT 25%

You can see I have accounted for all three possibilities with a total probability of 100%

If you now add up the PROBABILITIES (not possibilities) of rolling a head, 50%H plus 25%TH = 75%

thanks a lot. I got it.

Quote: GWAE

Also the term mathlete has been discussed as derogatory

By whom?

Quote: billryan

By whom?

https://wizardofvegas.com/forum/off-topic/gripes/24746-is-mathlete-an-offensive-term/#post508366

this is what the Wizard had to say about the word "mathlete" in his post at the above link:

Quote: Wizard

My opinion on the term "mathlete" is that it depends on the context. Used in a nice way, I would take it as a combination of compliment and term of endearment. Used in a nasty way, I would take it to mean nerdy and childish.

I agree with that and it was very obvious that I wasn't using the term in any kind of derogatory way. I was just asking for help.


GWAE's inane comment is taking political correctness to a ridiculous extreme

he didn't like my post - if he doesn't like a post he can just move on - he often complains about various members posts - as if he thinks the words posted here are carved in stone - that it's some kind of major faux pas to put up a post that he doesn't consider worthy



he needs to take a big ole chill pill

Quote: GWAE

Also the term mathlete has been discussed as derogatory

It may have been discussed, but it's not being used in a McFly, Where's my Homework sense I think it's a compliment, myself, though in my day it was Brainiac.