Here is part of a picture some of my friends posted from a recent high school reunion.
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.
- 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.
If the chance of one woman picking black is 1 out of 3. The probability of 2 women independently picking black would be:
And this expands to n women as:
Back to seven women. The probability of seven women independently randomly choosing to wear black would be:
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.
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:
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?
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 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.