Sunday, December 21, 2008

Why can't nine women have a baby in one month?

In this posting I plan on answering the famous question:

If it takes 9 months for a woman to have a baby, why can't 9 woman have a baby in one month?

If you don't like math, then you probably won't like my explanation.

So let's look at the general idea of splitting up a job. There are lovely quotes like
  • "helping hands make light work"
  • "the more the merrier"

The premise is that it takes 1/2 as long to do a job if 2 people are involved, 1/3 as long for 3 people, etc. The reality is that there is another factor called the "partitionability" or the ease with which a job can be split up.

There is another component to splitting up a job. It's the "communicability". In other words, how easy is it for everyone to communicate together when trying to get the job done. Get a scratch piece of paper and draw two dots. There is one line between the dots. For 3 dots, there are 3 lines. For 4 dots, 6 lines. For 5 dots, 10 lines. The equation for how many lines connect n dots is n(n-1)/2. Each dot represents a person and each line represents interaction between people in performing the job. Depending on the job, the "communicability" may be easier or more difficult.

When combining the partitioning with the communicability, we get the equation:



A plot of this equation looks like:



As you move from left to right, you are adding more people to help. At first you get the "helping hands" effect, and the time to do the job decreases. Then, you start running into the problem of having everyone know what everyone else is doing. That is where you start losing benefits from the "helping hands" effect. How many times have you worked on a project and somebody (maybe yourself) ended up frustrated and made one of these comments:
  • I don't know what is going on.

  • What am I supposed to be doing?

  • I thought that was my job!

  • As soon as I do something I find out someone else already did it.

For the specific case in the plot, there is actually a sweet spot where best size of a group to do the job is 5 to 8 people. This size depends on p and c.

It's also interesting to note that you are better off with one person doing the job than having 18 or more people. Of course there are other concerns here, like having the one person get hit by a bus. There is also another plot related to costs (I'll have to look it up later.)

Now we should be ready to apply this equation to a woman having a baby. Once again the question:

If it takes 9 months for a woman to have a baby, why can't 9 woman have a baby in one month?

First, if we look at the equation, the "partitionability" factor "p" is zero since you can't split up a baby between several women's wombs.

Second, the cost of communication factor "c" is infinite for the same reason. The resulting plot is here (c=0.1 instead of infinity):



The sweet spot is of course, one woman having a baby taking nine months.

The main message of this blog is to show how a scientist can use math to oversimplify reality and explain something that is already obvious. In reality it's very difficult to know the validity of assumptions when the scientific explanation is used for something less intuitive or obvious.

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