# If a gene is expressed at a level of 1/1200 compared to the average gene, how is probability 50:50 that we have a read mapped to it?

"To calculate the probability that a read will map to a specific gene, we can assume an average gene size of 4000 nt (100 M nt divided by 25,000 genes). At 30 M reads equivalent to 30× coverage, at single read 100 nt (or paired-end read 50 nt) length, we can expect a single read to map to the average expressed and length gene 4000 nt× 30 coverage/100 nt 1200 times. Thus, if the gene is expressed at a level of 1/1200 compared to the average gene, then we have a 50:50 probability to have a read map to it."

I understand that a single read would map to the average expressed 1200 times but I don't understand how the 50:50 probability is calculated. Please could you explain how this probability is determined.

I actually disagree with that.. I guess I write this down as a discussion.

The expected value for an average gene is 1200. For this gene at 1/1200 expression, you expect 1 read.

However, because of sampling, sometimes you get 0, 1 or 2. If we assume the sampling follows a poisson distribution, we can calculate the probability of getting 0, 1 , 2 ... reads:

plot(dpois(0:10,lambda=1),ylab="Probability",xlab="No of reads")


The probabilities are like:

data.frame(n=0:5,p=dpois(0:5,lambda=1))

n           p
1 0 0.367879441
2 1 0.367879441
3 2 0.183939721
4 3 0.061313240
5 4 0.015328310
6 5 0.003065662


The probability of getting exactly 0 or 1 read should be ~0.37. The probability of getting at least 1 read would be 1 - P(n=0) = 1- 0.367879441 = 0.6321206

• I guess the author just assumes uniform distribution instead of poisson. – haci Nov 19 at 14:20
• possible.. but why 50:50 ? – StupidWolf Nov 19 at 14:20
• I was curious about this question but didn't know what the distribution would be, so I left a very clumsy monte-carlo simulation running yesterday. After about 8,500 iterations I get exactly these values. – Jesse Nov 19 at 15:47
• One other thing: Could the author have taken the probability determined for the lambda=1200 value of getting >=1200 (about 50%) and incorrectly assumed it'd stay the same for lambda=1? – Jesse Nov 19 at 15:57
• There are a lot of back-of-the-envelope calculations in the original statement. I don't think 50:50 is supposed to be an exact number. The point is that you cannot be very confident in detecting that gene. – burger Nov 19 at 19:29