How unlikely is a cluster of pulmonary haemorrhage deaths in new born babies?

It is estimated that, in a typical large UK hospital, pulmonary haemorrhage deaths for new born babies is rare - it occurs between 1 to 3 times per 2000 births. The largest hospitals in the UK deliver about 690 births per month. What's the probability of seeing a cluster of 4 of these deaths in the same hospital in a single month? Using standard statistical assumptions (all explained and demonstrated in the 3-minute video below) the probability is about 0.7%, i.e. 1 in a 142 chance. So it is quite rare. But rare events happen all the time; in this case what we really need to ask is something like: how likely are we to see such a (monthly) cluster in a single hospital in the whole of the UK in over a year? The answer (assuming about 20 similar hospitals) is about 81%. In other words it would actually be surprising if we did not see such a cluster somewhere in the UK at least once a year. But what if we observed not 4, but 8 such deaths in one hospital in one month?

In this case the probability it happens in one hospital in one month is 0.00026% (1 in 385,000 chance). The probability it happens somewhere in the UK in a year is about 0.063%, (1 in 1,600 chance). So it is a genuinely unlikely 'event' which may well have a causal (non-random) explanation.

The video uses AgenaRisk software for calculations. The model can be downloaded here and run using the free trial version of AgenaRisk.