# Basic Microbial Bioreactor Contamination Probabilities Before my time at Cornell, I heard there was a professor who gave a grade based on the product of a test score and a lab grade.  If you got a:

• 10 on the test and a 9 in the lab, your score: 90.
•   9 on the test and a 9 in the lab, your score: 81.
•   8 on the test and a 7 in the lab, your score: 56 (ouch!)
When you get into mathematical situations where the final outcome depends on the product of two numbers being high, you're in a tough situation.

One time, I got tasked with figuring out what the probability of any natural disaster striking a biologics manufacturing plant given an estimate of the individual probabilities of said natural disaster.

So suppose:

5-year probability of disaster happening
DisasterProbability
Earthquake5%
Grass fire10%
Flood2%

The key to answer this question is thinking in terms of "not."

5-year probability of disaster NOT happening
DisasterProbability
Earthquake95%
Grass fire90%
Flood98%

So what's the probability that nothing happens?

0.95 x 0.90 x 0.98 = 0.838

Even though your probabilities of the individuals are in the 90% range, the probability that not any of them happen is in the low 80's. You have an 84% chance of nothing happening, which means the probability of something happening is 16%.

So the equation is thus:
1 - ( 1-p1 ) x (1-p2) x ... x (1-pN)
This is one of those huge mathematical bummers... the more things that can go wrong with your process, the success rate odds are stacked against you.

The same is exactly true with contaminations.

If a successful run depends on five separate operations. And the failure rates for those five operations are:
• p1 = 1%
• p2 = 1.5%
• p3 = 2%
• p4 = 2.5%
• p5 = 3%
Then success rate of the system is:

99% x 98.5% x 98% x 97.5% x 97% = 0.90%

Five measly steps each with failure rates less than 3% and your overall failure rate is 10%.

Next time you're troubleshooting your microbial bioreactor contaminations, think about this math. If your culture success rate is 10%, your status-quo aseptic practices can be executed >97% of the time and you can still contaminate 1 in 10 runs. 