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Someone once briefly explained to me why it is that chocolate chip cookies have 37% chocolate in them. To the best of my memory it has to do with the way trying to place dots in a circle in a random and scattered way behave, which turns out to be $1/e$ or $\approx37%$.

Chocolate Chip Cookie 37% 1/e

There are 2 levels of validating this:

  • The first is the theoretical side: Can you find a suitable definition for "random and scattered" in a circle that fits the use case and the $1/e$ behavior?

  • The second level is the practical. How does the size of the dots and volume impact this phenomenon? Are there real-life constraints that force it to be $1/e$?

Edit

http://sarcasticresonance.wordpress.com/2012/03/11/1e-do-not-lie/ This blogger validated the fact these do have 37% chocolate, and after contacting him he provided me with the following partial explanation, yet for this to become a full answer some conversion of the problem is needed and practical considerations to be taken.

Let us assume the following:

In the factory the manufacturing process start with a chunk of chocolate the size of a cookie, which is made up from a million chocolate particles. After that there are a million robotic arms. Each arm chooses a chocolate particle randomly and replaces it with cookie dough, unfortunately there is no synchronization between the arms, and its possible for few arm to switch the same chocolate particle. Its obvious that not all million of the particles will be switched but less, and therefore we will have some mix, the question is what will the ratio be.

Let us look on a specific particle, what is the probability of it being switched? hard to calculate directly since it may be chosen by some or all arms, but can be calculated throw elimination: an arm doesn't choose it if it happens to choose another particle.

That means:

$(N-1)/N = 1 - 1/N$

Is sum of all articles so the probability of no arm choosing it is N when:

$(1-1/N)^N$

And that makes the probability of an arm to do choose any particle:

$1 - (1-1/N)^N$

Well approach n to infinity, well use the know fact:

$(1-1/N)^N ----> 1/e$

We will get that in average:

$1-1/e = 0.63$

Which means 0.63 of the chocolate is being switch and there for 37% chocolate is left.

The question remains on the following points:

  • Can this be converted to scattering things in a circle? [this will make the next part easier]

  • Does this fit to real life constraints? or is it not?

edit 2

As requested and in reponse to the claim that 37% refers to the chocolate itself, iv'e added a picture of the back with the ingridients

back

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atntias
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    I feel like I must be missing something. The proportion of chocolate seems to me like it would be determined entirely by the recipe, and could easily range from 0% to 100%. – David J. Harris Jan 26 '14 at 06:51
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    you are correct that you can determine any range you want, but the question is there a practical or mathmatical reason for choosing 37%. –  Jan 26 '14 at 08:20
  • yes there is, apart from the titles on the boxes, this blogger went out to test it aswell, he is the one to give me the edited answer. http://sarcasticresonance.wordpress.com/2012/03/11/1e-do-not-lie/ –  Jan 26 '14 at 08:45
  • @David J. Harris: having a chocolate chip cookie implies that you need chocolate chips surrounded by dough. If the chips are melted together that is not a chocolate chip cookie anymore. Although it may still be delicious. – nico Jan 26 '14 at 09:18
  • I think I understand where the argument is coming from. Before baking, you start with a large amount of dough and a large number of chips, all of equal size. But when these get mixed in and baked, the chips melt somehwat and can merge with each other to form bigger chips. The blogger's argument shows that the proportion of chips/(chips+dough) tends to 1/e as the amount of each gets very large. It doesn't matter how much dough you start with because if you start with less dough, you just end up with bigger chips (continued) – Flounderer Jan 26 '14 at 09:45
  • To illustrate this, take a pack of playing cards. Treat the red cards as dough and the black as the initial chips. Shuffle thoroughly, then count how many groups of black cards separated by reds you get in the resulting mixture. For example, I got $13$ such groups. These are like the chips in the final cookie. In this case, the proportion chips/(chips+dough) in the cookie is $13/(13+26) = 0.33$. – Flounderer Jan 26 '14 at 09:48
  • @Glen_b I stand corrected. – atntias Jan 26 '14 at 10:27
  • @Flounderer: I you take a large number of ordinary decks of playing cards together, and shuffle them randomly, then with certainty in the result there will be (1) equally many red as black cards, and (2) up to a difference of at most $1$ equally many groups of red and of black cards. I cannot see by which feat of demagogy one could claim that the expected proportion of black (or red) cards or groups of cards is anything else than $1/2$, certainly not $1/e$. Even though it might be true (I did not check) that the expected number of groups is close to $n/e$ for $n$ cards. – Marc van Leeuwen Jan 26 '14 at 10:30
  • @Flounderer: If you started 2 chips the same size, and they merge to form one chip twice as large, you still have the same amount of chocolate. –  Jan 26 '14 at 10:56
  • @Hurkyl i agree, yet that point still stands at the heart of the problem, the need for 2 chunks of chocolate to be separate is behind the "scattered behavior", and seems to be the real world constrain that can tie the mathematics and the reality, yet the whole picture isn't clear. – atntias Jan 26 '14 at 11:05
  • It's because $37$ percent is basically $40$ percent, which is pretty much the same as $50$ percent, meaning that half of my cookie is chocolate chips! Oh man! – MT_ Jan 26 '14 at 19:34
  • @Marc van Leeuwen the feat of demagogy is to count the red cards individually but count the black cards in batches. @ Hurkyl I agree that it doesn't make sense that they would measure the proportion of chips in the way I suggested. What else could $37%$ mean but $37%$ of the mass? – Flounderer Jan 26 '14 at 20:50
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    @Flounderer - how did you get 13 groups of black cards and 26 groups of red cards? Pretty sure that's impossabible. You could only get 12, 13, or 14 groups of red me thinks. – probabilityislogic Jan 26 '14 at 21:52
  • @probabilityislogic I was counting the red cards individually but counting the black cards in batches – Flounderer Jan 26 '14 at 23:24
  • I hate to be a party pooper but where does the cooking come into the description above. When the dough is cooked it is transformed and changes. Are these machines moving the dough particles before or after the dough is cooked? How does this explanation account for the changes in the chocolate to dough ratio after the loss of water weight during cooking? – dfc Jan 27 '14 at 18:24
  • 37% (that is, 1/e) isn’t arbitrary; it’s the optimal depth of search. This may be just a coincidence, or perhaps this physical situation actually models this phenomenon. https://en.wikipedia.org/wiki/Secretary_problem –  Jan 07 '21 at 16:26

3 Answers3

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37% refers to the amount of dry cocoa solids in the chocolate chips. These percentages are how chocolate is sold.[1][2] In the US and EU you need more than 35% cocoa solids in order to refer to your product as semisweet chocolate and chocolate (respectively).[3] The chocolate bar recommeneded in this review[4] as "splurge worthy" is labeled 61% dark chocolate. This is not because the bar is 61% chocolate and 39% air/other. It is a 61% chocolate bar because the chocolate contains 61% dry cocoa solids.

[1] http://www.thestoryofchocolate.com/Savor/content.cfm?ItemNumber=3454&navItemNumber=3376

[2] http://www.21food.com/products/brick---2-lbs---milk-37-51823.html

[3] https://en.wikipedia.org/wiki/Types_of_chocolate#Classification

[4] http://www.cookinglight.com/cooking-101/essential-ingredients/taste-test-chocolate-00400000058842/

dfc
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  • intresting point, yet im not convinced, in this brand of chocolate chip cookie the cocholate that exists is 2/3 dark chocolate and a 1/3 milk chocolate, and i refer you again to to show that the cookie dough ratio is infact 1/e http://sarcasticresonance.wordpress.com/2012/03/11/1e-do-not-lie/ – atntias Jan 26 '14 at 19:31
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    Yes one Lets get one thing straight that blog did not identify the cookie dough to chocolate ratio. That one shot non scientific investigation was done with baked cookies. Why do you think that the combination of semisweet and milk chocolate does not result in a 37% chocolate? Do you have a picture of the package with those chocolate percentages? – dfc Jan 26 '14 at 19:48
  • the blogger uses the same brand of cookie and has a picture of the box there (there is Hebrew text on it but its the same cookie trust me i eat them all the time), he also does a pretty good job testing it, and shows there is cookie dough ratio very close to 37%. be precise, im getting the feel your not reading it trough, i edited the post to include a picture of the ingredients in the back, and if you convince me there is a reason il test the ratio my self. – atntias Jan 27 '14 at 05:38
  • Notice the cocoa mass and cocoa butter in the ingredients list? Can I also "convince you" that you are not talking about the "dough ratio"? Dough goes into the oven and cookies come out. That blog tested cookies not dough. – dfc Jan 27 '14 at 18:21
  • am i missing something? he took the overall weight of the cookies then dissolved the "dough", that leaves you with only the chocolate, then calculated overall weight minus chocolate weight which is the dough weight, then do the ratio. what did i miss? – atntias Jan 27 '14 at 18:32
  • ah you mean semantics, ok fine, but not constrcutive in any way. – atntias Jan 27 '14 at 18:34
  • It is constructive if you consider your initial presentation. The dough changes when it is cooked. Are you million arms moving dough? If so it seems that the cooking process would alter the ratio. Ignoring my comment on the cocoa mass/butter is not constructive. – dfc Jan 27 '14 at 19:30
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Businesses prefer round or familiar numbers -to make it easier to compute by mind. So I am willing to bet that the the company would have been more inclined to have, say, a round 40% content (or 8 gr in the 20 gr overall that each cookie weighs according to the blogger), rather than $1/e$. Or even, 33% (which is familiar in the sense that is translated as $1/3$ or a $2:1$ ratio). So why did they end up with $1/e$? I conjecture it has to do with production cost considerations.

This is a consumer's product, produced in large quantities. For legal and commercial reasons, it needs to be as homogeneous as possible. One way to achieve this would be to create each cookie one-by-one. It doesn't take much thought to realize that this would be a very costly process...

So cost-considerations mean that we do not create the dough-chocolate mix cookie-per-cookie, but we create one large mass of dough-chocolate and then we separate it in parcels of same weight and send to bake, in order to arrive at the ~20gr ready cookie. Given this production process, in order to achieve homogeneity among final cookies, we need to have the chocolate chips as homogeneously distributed as possible in the dough mass. Therefore the machines in the factory need to "mimic" a "random selection" process (selection of the 3D point that each chocolate chip will occupy inside the dough mass), so that they are scattered ~uniformly in this mass.

Now the cookie dough is not dry, but watered. So it is less messy (and so less costly) to move around the chocolate chips in transfer-belts instead. Then it looks like they prepare the cookie dough in some kind of large containers, then pour in the chocolate chips, then "shake" (the correct English word escapes me) in order to make the mix as homogeneous as possible... And here comes another cost issue: how long do they need to shake the mix in order for it to become homogeneous? The longer it takes, the costlier it is. So a cost-sensitive business would opt for the minimum possible amount of "shaking time", that would guarantee 3D-homogeneity of the mix as regards the distribution of the chocolate chips.

And this is the answer "why $1/e$": if the weight proportions of the total dough mass and the total chocolate chips are such that conform with the theoretical end-result of the theoretical random selection process already described in the question, it will take the minimum amount of time to effectively deterministically mimic this random selection process -i.e. to make the whole mix homogeneous.

As to why "if this is so, why not everybody is doing it?", the answer has to do with the fact that not all businesses are primarily cost-driven (a firm may want to have a 50% chocolate content and charge a higher price), and that history matters -a cookie may have traditionally contained 20% chocolate, and be commercially successful, and they didn't want to jeopardize that.

Alecos Papadopoulos
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    i agree with all you said like your way of presenting it, it does most of the logical work, yet i feel you haven't nailed it all the way and il explain: as i see it, at the heart of the problem there are 2 constrains, the first is that you don't want chocolate chips to merge into a chunk and look uneven, here the less chocolate the better (less chance for the chocolate to bond), the other is the desire to have a large amount of chocolate in the cookie. to these add the facts you mentioned about automation. the question is: what mathematical phenomena causes these constrains to meet at 1/e. – atntias Jan 26 '14 at 12:26
  • As for the first issue: one has to determine how large each piece of chocolate should be, given the optimal cost that dictates a $1/e$ proportion in weight. So first we decide on $1/e$ and then we determine the size of each chocolate chip. For the second, I do not agree that any such "desire" exists in a business approach to the matter: we are only in it for the money here -and a "large amount of chocolate in the cookie" does not necessarily bring maximum profits. Mind you, I agree with you that a complete answer should be totally mathematical -but this would not be an answer, but a paper. – Alecos Papadopoulos Jan 26 '14 at 12:57
  • i don't want to get too philosophical but, its obvious that if were selling chocolate chip cookie we want to have some chocolate in them, and yes the fact it says all over the packaged 37% shows that there is a branding effort to show there is a substantial amount of chocolate here, the efficiency consideration are an addition to that, as to your claim it would take a paper I'm not sure, I'm not asking to explain the very nature of the phenomena itself rather then reference to its description and exsistnace, from there it likely that the paper exists already. – atntias Jan 26 '14 at 13:05
  • ''"shake" (the correct English word escapes me)'' Agitate? –  Jan 27 '14 at 01:38
  • I think to have some claim to answer the question, one should give at least one relation with a mathematical problem whose outcome is (approximately) $1/e$. I see no mathematical modelling whatsoever in this answer. – Marc van Leeuwen Jan 27 '14 at 06:24
  • @MarcvanLeeuwen Mathematical modelling, there is, mathematical notation, there isn't. I agree with you that the latter should also be present in my answer for it to be complete. The mathematical problem whose answer is $1/e$ is already stated in the question - I don't believe I had to replicate it in my answer. – Alecos Papadopoulos Jan 27 '14 at 07:48
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    @AlecosPapadopoulos: Your answer makes no connection at all to the model of production (millions of robotic arms, which is by the way utterly ridiculous) mentioned in the question. So I cannot see how you would get any justification for the number $1/e$ from that model. – Marc van Leeuwen Jan 27 '14 at 08:03
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Suppose we are making $c$ cookies (where $c$ is a large number). For simplicity, let's approximate by assuming each cookie has $k$ discrete locations where chocolate chips might go (where $k$ is a positive integer). Then there are a total of $n = ck$ locations where chocolate chips might go. So we start with $n$ chocolate chips and randomly throw them down. Each one independently lands in one of the $n$ locations. If two or more land in the same location, they all bounce into the garbage.

So at the end of this process, a location has a single chocolate chip if and only if exactly one chip landed there. Now given any particular location (of the $n$ possible ones), the probability that a particular chocolote chip attempts to land there is $q = 1/n$. The probability that there is a chocolate chip there after this process is done is: $$ nq(1-q)^{n-1} = (1-1/n)^{n-1} $$ When $n$ gets large (so that $n\rightarrow\infty$) we get: $$ (1-1/n)^{n-1} \rightarrow 1/e $$ and so the fraction of chocolate chips that are not thrown away converges to $1/e \approx 0.367879$.

Michael
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