Formal analysis of birth control aggregation

The quantity we are interested in is, for some sample of 100 couples \(c_{i},i=1\ldots 100\),

where \(I(\cdot)\) is an indicator variable. We can exchange the sum and the expectation (expectation is a linear operator) and this does not rely on the independence of the couples or their outcomes:

Note that here the expectation is still with respect to the distribution for couple \(c_i\), which is why I’ve subscripted it. Now, an expectation of an indicator is just a probability, so we can simplify.

The next step is where I diverge from the NYT. They (implicitly) assume that \(\Pr_i(\cdot) = \Pr_1(\cdot)\) for all \(i\). That allows another simplification:

I’m challenging that assumption, since it would be rather odd to sample 100 couples and find they all had identical distributions. Without knowing something of the distributions, you have to leave it at the penultimate step above. In my numerical example, I’m imposing a mixture of two types with distinct distributions, which would still be consistent with the observed group single-year effectiveness, but which results in quite a different answer to the question above.

To actually calculate things we can treat the probability as a single Bernoulli trial lasting 10 years, or we can break it down into a sequence of 1-year Bernoulli trials by imposing a (potentially reasonable) independence assumption. Actually, this doesn’t matter for my argument - the error has already happened by this point.