... closer to 1:30.

an accumulator on five independent unrelated 1:30 events is 1: 24,300,000

You've calculated 5

^{30}=24,300,000, or the total number of ways that 5 people can die in a 30 day period with a chance of 1:30 to die each day.

However, you have over estimated the number significantly. For those 5 individuals, there would be 30 different combinations that they could die on the same day (all on the first of the month, all on the second of the month, and etc...).

So, considering only 5 individuals, you've over-estimated by a factor of 30, and our actual number would be:

24,300,000/30 = 1:810,000. Then consider 12 months, or 365 days a year. So, let's divide that again by 12, and we get: 1:67,000 in any year. If one considers the 16,100 US nursing homes, we end up with a 1:4 chance this happening in the USA in any year.

Let's consider nursing homes.

In the USA, there are about 16,100 nursing homes with about 1,492,200 residents, or about 93 residents per nursing home, or about 0.5% of our population.

So, rather than doing the statistics on only 5 patients, you need to consider all 93 patients per nursing home, and the thousands of nursing homes in any country.

I got lost on the statistics considering that as beds are emptied, they are refilled, length of stay, severity of the condition, hospice patients, and etc.

My guess is that these 5 patients has varying underlying conditions that made them unable to stay at home. Emphysema, CHF, Liver or Kidney failure, Cancer, Broken Bones, Mental deterioration, strokes and vascular disease, etc. They may have had DNR orders, which can also limit the medications and treatments that a patient will receive.

Then, on top of that some of them might have caught a common infectious disease (cold, flu, what-not), that sent them over the top.

So, one issue would be whether the nursing home had adequate infection control.

However, this is likely little more that a slight statistical anomaly.

Consider our stats above. If a nursing home had 90 patients, each with an expected lifetime of less than 30 days, then one would expect about 3 to die a day, or 6 to die in almost every 2-day period. That would be a bit high for a death rate, but not impossible for some hospice patient populations. I believe my grandfather's stay in a nursing home was about 30 days from admission to death, and we got home-health care and hospice for my grandmother about 2 weeks before her death.