For an optimization problem related to media planning I needed to evaluate the probability that a multinomial variate assume the value zero. This can be written as:
However it was suggested to use the following expression:
I did not see immediately that these things are approximately the same. With a little bit of effort:
where the last step is based on a Taylor approximation:
To be precise:
This approximation in evaluating the reach for some advertisement plans seem to work very well.