Drake and the Back of the Envelope

One of the many handy things about the math of probability is that we can simply stack probabilities of individual events to get the overall probability for the scenario. For example, the probability of a coin landing on heads is 0.5 and the probability of three heads in a row is 0.5 * 0.5 * 0.5. We can use this same principle to break down seemingly complex problems into smaller, more-manageable chunks.

A great example of this is Frank Drake’s eponymous equation. This equation is supposed to calculate the probability of alien civilizations in our galaxy. Of course we have no way of knowing this number, but the equations gives us several components that we can at least take a guess at.

  • the average rate of star formation per year in our galaxy
  • the fraction of those stars that have planets
  • the average number of planets that can potentially support life per star that has planets
  • the fraction of the above that actually go on to develop planetary life at some point
  • the fraction of the above that actually go on to develop intelligent life
  • the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
  • the length of time for which such civilizations release detectable signals into space

If you want to play around with these numbers, the BBC has an interactive infographic about the equation.

Doing this type of rough estimate is often referred to as a “back of the envelope” estimate–the idea being that you are just jotting down numbers of a piece of scrap paper to get a quick answer. While inexact, this is actually a very useful exercise that when applied properly will give you a better idea of an actual number or probability.

Consider a meeting where you’re asked how many customers cancelled their accounts in 2012. You don’t have that number with you, but can you make a guess? It’s more likely that you know the number of customers who cancelled last week. If five customers cancelled and you don’t feel that number is particularly low or high, multiply that by the number of weeks in the year to get 260. Next you might want to knock off a few for growth if you company has more customers this year than last. So take off 3-5% and you can report an estimate of around 250. This is a lot better than “I don’t have any idea” or pulling a number out of no where.

Have you used this method before? What types of problems do you encounter where it is useful?

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