# How to compute the chance of failing to detect a gene given the detection limit of a protocol

In Shapiro et al., when discussing about loss of molecules as source of error in single-cell sequencing, it is written that:

Another source of error is losses, which can be severe. The detection limit of published protocols is $5$–$10$ molecules of mRNA. If, as seems likely, the limit of detection is primarily determined by losses during sample preparation, this would indicate that $80$–$90\%$ of mRNA was lost. Or, to put it the other way around, a $90\%$ loss leads to an approximately $50\%$ chance of failing to detect a gene that is expressed at a level of seven mRNA molecules (from the binomial distribution).

How is this probability computed using the binomial distribution? I thought that $90\%$ loss corresponds to $5$ detected molecules, and I assume that $k=7$ for the binomial calculation, but I am unable to go further.

A 90% loss can be rephrased as a 10% chance of detecting anything. So what we want to find is the probability of detecting 0 molecules, when we start with 7 and have 10% probability of success. Once can do that in R as follows:

> pbinom(0, 7, 0.1)
0.4782969


So ~50%, as they stated. I suspect that part of the confusion arises from the fact that the detection limit is due to lowly-expressed genes/transcript being heavily affected by this loss. So the probability of detecting a single molecule out of 4 original molecules (assuming 90% mRNA loss) is 34%, for 3 molecules it's 27%, for 2 it's 19% and for 1 it's 10%. I think the threshold of 5 is mentioned more because it's a nice round number than there's anything particularly different in detecting a gene with 4 vs. 5 molecules (34 vs. 41% probability).

• I was reading again your answer to understand the first sentence. If I understood correctly, the mapping from $90\%$ loss to $10\%$ chance happens computing the probability of detecting a single molecule out of $1$ original molecule. Right?
– gc5
Dec 19 '17 at 20:41
• Correct, since there's a 90% loss of all signal, there's a 10% probability of detecting any given molecule. Dec 19 '17 at 23:52
• Detecting zero molecules out of 7 can be rephrased as failing to detect a molecule, 7 times independently. The probability of this is 0.9 to the power of 7. Python tells me (0.9 ** 7) that this is 0.4782969. That's probably what the modelization in terms of binomial distribution means, but to me it is much easier to understand in terms of this very basic probability calculation.
– bli
Dec 20 '17 at 13:25