Dot Physics

Here is part of a picture some of my friends posted from a recent high school reunion.

i-a93ab66e8fc2c6d79e67484791d6434b-2010-07-23_black_dress.jpg

It may be hard to tell, but this is part of a picture of 7 females all wearing black. I just wanted to show you that they were indeed wearing black without giving away anymore details. If you are one of these people and you want your whole picture included, I will be happy to make that change and include your face.

Anyway, my first comment was: “Wow, everyone is wearing black. Was this a planned event or was black part of the dress code?” The response was that it was just pure chance that all the women were wearing black. ALL CHANCE? All chance you say? That seems unlikely – but let me crunch some numbers just to be sure.

Assumptions

  • As my friend claimed – there was no plan. This suggests that she believes each woman independently decided to wear a black dress. I am glad, this makes things easier.
  • Here is the big one. If a woman was picking out an outfit to wear, what is the chance that outfit would be black? I originally guessed 1 out of 4, but maybe that should be 1 out of 3. I am going with 1 out of 3 because black is a slimming color and black is the new black.

Calculation

If the chance of one woman picking black is 1 out of 3. The probability of 2 women independently picking black would be:

i-4e1d723992d393cecd3628061e2e74d4-2010-07-26_la_te_xi_t_1_2.jpg

And this expands to n women as:

i-1f5810419c126f390d4706809a47b0f8-2010-07-26_la_te_xi_t_1_3.jpg

Back to seven women. The probability of seven women independently randomly choosing to wear black would be:

i-d60d2bd8b5dc68bfa257e5f36a01156b-2010-07-26_la_te_xi_t_1_4.jpg

Right there you have your answer. If the women are independently choosing their outfits and if they choose black 1 out of 3 times then there is a very small probability that they all chose black. It could happen, but it is not very likely.

Simulation

I can’t stop there, I just can’t. How about I simulate 7 women meeting at a party. For simplicity, let me say that a woman randomly chooses 1 of three colors: black (B), color A (A), or color B (B). I know it is actually more complicated than that. It is probably something like 9 choices of outfit, but 3 are black – but the result is the same.

Suppose I went to 20 meetings where these 7 women randomly chose a dress. I then count how many were wearing black. Here is what that might look like:

i-19b58e5524f84070407e482ac1501467-2010-07-26_figure_1.jpg

These 20 random meetings, not once were all the women wearing black (not even 5 or 6 of the 7 wearing black). (note that if I re-run the simulation, it is possible to see this happen once or even more than once). What if I went to 500 events?

i-8d18a63b723fd4cbd33735b820c995c8-2010-07-26_figure_1_1.jpg

Again, not all were wearing black. Only 2 out of 500 had 6 out of 7 wearing black. Ok. One more graph. How about 5000 events?

i-dbe02c752ed6341a4ffca236cabdf523-2010-07-26_figure_1_2.jpg

I know you can’t tell, but actually 2 of these 5000 events had all seven women wearing black (it is just such a small number compared to the other possibilities). Boy, I sure would be tired going to that many events.

The next step

Probably the next step would be to go out to some events and count how many of the women are wearing black. I am not going to do this.

Comments

  1. #1 stripey_cat
    July 26, 2010

    I think you’re hugely underestimating the likelihood of women choosing to wear black. Yes, I may have three evening dresses in different colours, but the black will see the heaviest wear (not least because people will remember the coloured dresses much better than the plain black; so if you don’t want to be seen to be wearing the same dress often, black is a good choice). Additionally, black is seen as a safe social choice. So for an event, like a reunion, where you’re a bit nervous of the impression you will make, you’re more likely than normal to choose black.

    Meetings are a little different: the range of safe choices expands to include charcoal, navy, dark brown, or pinstripe. Also there is less stigma associated with wearing the same suit multiple times. So suit colours are more likely to be chosen to flatter the wearer or to follow fashion.

  2. #2 the Rising Jurist
    July 26, 2010

    I agree with stripey_cat. Black is not only safe, it’s the formal color of choice (indeed, for men, it is the only acceptable color for an evening function). Though women are not expected to adhere to this color rule, the fact remains that it is a risk-free choice. There is a reason the “little black dress” is considered a must-own item.

  3. #3 Flavin
    July 26, 2010

    If this picture was taken at a reunion, there were likely more than seven women there. If there are seven women there, the likelihood of all seven wearing black is low, but if there are twenty or thirty women there, the likelihood of any seven wearing black goes up.

  4. #4 Alex
    July 26, 2010

    It looks like you are calculating the probability that *these seven friends* all chose to wear black. You could also ask ‘what is the probability that at least 7 women at the reunion wore black?’ If there were say 50-100 women at the reunion then it is a near certainty that at least 7 of them wore black. The chance that *some group of seven friends* all chose to wear black is probably not that unlikely. This seems like what you would find if you took your next step (assuming you went to events with more than 7 women).

  5. #5 Dave X
    July 26, 2010

    Or count google images… Counting one of the “10 year class reunion cocktail” pictures looks like 10 of 16, and (10/16)^7=3.7% isn’t terribly extreme.

    There’s also a lot of opportunities for selection bias – choosing to pose in the picture, taking the picture, posting the picture, and blogging about it.

    Maybe revising the big assumption would be good. A 5% chance of this particular outcome might mean a 90% (=(5%)^(1/7)) chance of wearing a black, if you think it unlikely your friend would lie to you.

  6. #6 wrpd
    July 26, 2010

    Are you sure you weren’t at a funeral?

  7. #7 CalcDave
    July 26, 2010

    THIS is exactly why “real life math” scares me. Sure, you can calculate theoretical probability all you want, but there are SO MANY variables that you would need to account for to get it even close to “accurate:” the 7 women in the picture are friends, so they probably think-alike and may have similar wardrobes; where is the reunion taking place (at the alma mater’s gym? some formal restaurant? a ballroom?)?; do the women have dates/husbands that they want to match?; etc etc etc

    THEN there’s also all the “leg-work” of getting real-world numbers for those variables.

    I guess that’s my slight-autism showing that this sort of stuff scares me. I’ll stick to the “theoretical stuff.”

  8. #8 OmegaMom
    July 26, 2010

    Add in the fact that it was obviously a dinner or cocktail party (those dresses are *dressy*, not casual).

    The “conspiracy” is one of the fashion world: The “Little Black Dress” is a party-going cliche for women. Take a look at the dress section online of any higher-quality department store (Nordstrom’s, Macy’s, etc.); the LBD usually has an entire section all by itself. The one at Nordstrom’s right now ( http://shop.nordstrom.com/c/6014148/0~2376776~2374327~2374331~6014148 ) has *ten pages* of LBDs to choose from.

    Now, if they had all shown up wearing *red* dresses, or another color, I would have been much more willing to accept your thesis and conclusion.

  9. #9 Katherine
    July 26, 2010

    Red dresses are overrepresented at other types of events. Every dance competition I’ve been to, over half the women will be wearing red dresses in an attempt to stand out and look sexy. What actually ends up standing out is ANY other colour. There, I’d be surprised if more than half were wearing black dresses.

  10. #10 Paul
    July 26, 2010

    Two thoughts — both are me nagging for more data, sorry! ;)

    1. Do you have other photos of women at the event? Can you count up black vs. nonblack outfits?

    2. If so, it would be interesting to estimate the individual probability of wearing black (instead of guessing) and then estimating the probability of getting that picture given the number of photos taken?

  11. #11 CCPhysicist
    July 27, 2010

    One other critical bit of info would be WHICH reunion, the stated dress code and location (sure does not look “casual”). I don’t recall seeing a single hot color cocktail dress at my 20th or 25th, with one exception: the much younger date of a recently divorced guy.

    But you gave me something to look for at an upcoming reunion!

  12. #12 rob
    July 27, 2010

    looks like a Poisson Distribution.

    knowing how many women were at the party and probability of wearing a black dress, you ought to be able to calculate the odds of 7 women wearing black.

  13. #13 Ania
    July 27, 2010

    It is how lotto works.
    You will never guess what numbers will be, although some people are winning.

  14. #14 Jon
    July 27, 2010

    I think what’s missing here is the hypothesis – or at least, it wasn’t explicitly stated. Your “first comment” sort of points to what I think you’re asking – is this really pure chance? The very low chance of this happening by pure chance, combined with the many ways other commenters have come up with to explain how a choice would be biased towards black, makes me tempted to reject the hypothesis that this happened by pure chance. Instead I’d argue that the choice of what to wear by each woman was not actually independent. If they attended the same high school, are friends, and all attended the reunion, then they’re probably part of the same culture. And despite my lack of a fashion sense I’d bet that fashion has a cultural basis.

    I don’t think I could conclude it wasn’t random though. After all, how many high school reunions happen each day/year/whatever in the US? I’d guess very many, and we’ve only got a sample size of 1. So it’s maybe possible that at least one reunion had a group of 7 friends who all wore black purely by chance, just that the reunions mostly did not have that. Maybe you were “lucky” to have this photo come from the rare black dress reunion.

    I not sure I’d try to explain this by the size of the reunion itself though. We’re asking about the chance of 7 specific friends all wearing black. The chance of any group of 7 friends or of any 7 women at a reunion are different situations with different probabilities.

    It’s been a while since I took statistics and sadly I haven’t had much chance to use what I learned since then, so maybe I’ve gotten everything wrong here. But this reminds me in a way of the idea of using a z or other statistical test in conjunction with a null hypothesis. I bet with a little tweaking this post could be used to teach that.

  15. #15 Alan
    August 2, 2010

    Hello I’m Alan from England. I believe black Dresses are good and elegant, especially if you know how to match it with some jewelry.
    If you have time please check out also my Dresses collection on my site. I am pretty sure you will like them.Thank you so much.:)
    http://bit.ly/HxgPt
    Alla wonder =)

  16. #16 Jane
    August 5, 2010

    I think you are totally over looking the fact that black is a very popular color in clothing. Everyone wants to own a little black dress and they all save them for special occassions to wear them. I think it was more of a coincidence rather than some scheme.

  17. #17 Ricardo
    August 7, 2010

    Alex has a point: How many women there were in the party? In fact, you don’t require 50-100 to have a good chance that 7 will wear black.

    With 20 women, and with probability of wearing black equal to 1/3, the probability of at least 7 of them wearing black is already 0.52. This can be calculated using the binomial distribution.

  18. #18 air max 2009
    September 6, 2010

    Madalena, parabéns pelo presente! E parabéns à Carla, pois o trabalho dela é muito bonito!
    A amizade é cada dia mais rara de se conquistar! E, para nós que temos um contacto “físico” com as pessoas muito pequeno resta o mundo virtual. Conheci meu marido nesse mundo e já estamos juntos há 7 anos! Conheci pessoas lindas, em que tenho muita amizade! Adoro vir aqui visitar seu blog, pois senti uma empatia grande contigo! E é por isso que volto sempre!
    Uma boa semana e um beijo!

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