A client asked me recently a fun probability question, which revolved around figuring out the probability of success of a research program. In a simplified form, here is the problem: imagine that you have multiple labs, each developing products which have independent probabilities of succeeding – what is the probability of more than a certain number of products being eventually successful?
Let’s illustrate on a simple example. Product A has a 30% probability of success, and product B a 60% probability of success. Combining these into a probability tree, we work out that there is an 18% chance of having 2 products successful, 18% + 12 % + 42% = 72% chance of having 1 or more products succeed, and 28% chances of a total failure.
It’s not a very complicated theoretical problem. Practically, however, when the number of products increases, the number of outcomes becomes large, fairly fast – and working out every single combination by hand is extremely tedious.
Fortunately, using a simple trick, we can generate these combinations with minimal effort. The representation of integers in base 2 is a decomposition in powers of 2, resulting in a unique sequence of 0 and 1. In our simplified example, if we consider the numbers 0, 1, 2 and 3, their decomposition is
0 = 0 x 2^2 + 0 x 2^1 –> 00
1 = 0 x 2^2 + 1 ^ 2^1 –> 01
2 = 1 x 2^2 + 0 x 2^1 –> 10
3 = 1 x 2^2 + 1 x 2^2 –> 11
As a result, if if consider a 1 to encode the success of a product, and a 0 its failure, the binary representation of integers from 0 to 3 gives us all possible outcomes for our two-products scenario.