# Calculate the average rate of success

It sounds like a simple problem but I am unsure about the methodology I should follow. A company developed processor with a new innovative way and they tested the processors to see if they pass the basic quality assurance test. They tested 100 processors and they want to know the average rate of the processors that pass the test. Also, they want to develop a model that can calculate the rate and infer it from a set of data collected.

It sounds like a simple problem where I have to find the success rate. If that's the case I would simply find the number of processors that passed the test and divide it by the total number of process that were used in the test. Is that right? I don't understand what kind of model I should develop for that.

If you are asking about the average rate of success based on $n$ trials and $m$ successes, it would be $m/n$, as you point out.

If you are looking for a specific probability model for this experiment, check out the Binomial distribution. If you have $N$ independent trials (in this case a trial is testing an individual processor for success or failure) and $p$ probability of success for each processor, then the average, or mean, success rate is $Np$, which is what you are looking for. See more here.

For example, if you knew out of the 100 tests, that 60 passed, your average rate of success is 60%, which is your $p$. Now you can make inferences on the next test. Say the next time you ran the experiment you tested 30 processors. What is the probability that 20 will pass?

from scipy.stats import binom
prob_success = 0.6
print binom.pmf(20, 30, prob_success)

# probability of 20 successes in 30 trials given an average success rate of 60%
0.11518537647948386

• Thanks for you help! In my case I have a dataset with the number of tests and the results(1 for pass and 0 for fail), in this case I can find how many passed and follow the approach you suggest. I could also suggest the solution when we only know the probability of success, just to cover both cases. – Dr.Fykos Jun 9 '15 at 18:18
• No problem. FYI - each test is called a Bernoulli process, which only has outcomes 1 or 0 (pass or fail). When you run multiple independent tests, each with the same probability of passing or failing, such as your example of 100 processors then you get a Binomial distribution. – ilanman Jun 9 '15 at 18:25