Today I was listening to the news, and some of you may have heard but Sydney is in Lockdown now, because of a Covid Outbreak, which is up to just over 100 cases.

Its been about 3 months or so since there was a cluster of local cases, and so this meant that Sydney was presumably COVID free. Which makes for a good analysis of how COVID can spread and how effective vaccines are.

# The party where it happened

In the news today, the NSW health minister described the following situation.

There was a party with 30 attendees, one of those attendees was infected with Covid and had been linked to a previous exposure site.

In the days following the party, there were 24 infections, and 6 people that didn't contract COVID.

There were 6 Vaccinated attendees, and 0 were infected.

there were 24 non-vaccinated attendees and all 24 were infected.

Obviously the statistics speaks for itself, the vaccine is highly effective, and this is confirmed by intuitively looking at the statistics, but how sure can we be? I thought I would go through how you would analyse this problem statistically.

# Statistical analysis

One way to analyse this situation statistically is to assume that the vaccine makes no difference, and work out the probability of having the same outcome that occurred.

So to do this, we assume the probability of being infected is 24/30 or 80%.

This means the probability of not being infected is 20%.

Hence, to get all 6 vaccinated not infected, it would have a probability of 20% times itself 6 times, or simply 0.2^6 = 0.006%

**0.006% is equivalent to one chance in 15,625.**

There is a 0.006% chance of the vaccine not contributing to the outcome, and just random chance resulted in that outcome. Its pretty certain even in this small sample the vaccine makes a difference.

We can also test the other side as well, lets assume that there is an 80% chance of being infected, what is the chance that all 24 non-vaccinated individuals get infected?

So for all 24 to be infected, that would mean the probability would be 0.8^24 = 0.47%. That's saying, if the rate of infection is unaffected by the vaccine then there is a 0.47% chance random chance resulted in all the non-vaccinated individuals to be impacted.

Now I know I have taken some short cuts in allowing for sampling errors etc, and assuming the probability of infection is the sample mean as opposed to a population mean, but I think the simplistic way I have described it above is close enough to reality, but much easier to understand.

# Variants and Vaccine types

Whilst not mentioned by the health minister, the vaccinated individuals were all health workers (as they had priority) and most health workers were vaccinated with the Pfizer vaccine.

What is certain is that the COVID variant at the party was genomically sequenced to be the delta variant, the new highly contagious variant spreading throughout the world first seen in India.

A news article referring to this can be accessed form the following link:

https://www.abc.net.au/news/2021-06-28/vaccinated-attendees-west-hoxton-birthday-party-avoid-covid-19/100249612

Now this post isn't meant to promote vaccinations, I just saw it as a unique opportunity to explore statistically something that was in the news.

Posted with STEMGeeks