I need help to solve these 3 questions below:

# A PCR test for SARS-Cov2

PCR tests can be made on single persons - one at a time - or they can be made in pools where samples for several persons are mixed before testing.

The idea behind pooling - proposed but not operational in current practices - is powerful.

If your goal is to test 10.000 persons a day (sounds familiar ?) you can do the following:

### Single testing

You do 10.000 single tests (one PCR reaction pr. sample) - that is expensive and cumbersome.

### Pooled testing

You could say "Let's do 1000 tests on 10 pools (where 1 pool contains 10 samples)".

If a pool is "positive" for the virus then you test individually all 10 samples in the pool. If the number of carriers is initially rare it magnifies your screening power 10 fold!

## False negative in pooled samples

We then turn to what happens when samples from different persones are pooled together before PCR testing.

We still want to calculate the probability that the RNA from a carrier is being missed by the initial PCR reaction mix (a False Negative).

The pooling dilutes the contribution of each individual sample to the PCR reaction mix.

We assume perfect pooling: every individual contributes the same amount to the PCR mix.

So for instance if 24 out 48 individuals are carriers, we use a Poisson distribution with mean lambda = 25 to model the number of viral molecules that make it to the PCR mix.

If at least 1 physical viral RNA copy is present in the PCR mix, it will be amplified and "bingo!" the test for this pool will be positive (as it should).

Imagine now that we use a pool of 48 individuals. We assume perfect pooling and focus on the situation where only one individual is a carrier, the 47 others are not infected.

>Q3.2 What is the probability for a negative result when testing a pool with 1 carrier mixed with 47 non carriers ?

## How to work around the false negative problem

To ensure reproducibility, each tested individual is put in three different pools.

Now we assume that, because they are very few carriers, that a carrier individual will still be the only carrier in all three independent pools.

So now we will miss this individual only if all three pools are false negatives.

>Q3.3 What is the probability for a negative result in all 3 pools, each containing1 carrier mixed with 47 other non carriers ?

>Q3.4 What is the minimum of replicated pools to use if you want to ensure that the probability of detecting the single carrier is >99% ?