Application of binomial distribution

Hello all, this is my first post. I am a practicing engineer facing a dilemma. Our resident statistician is out for the MONTH, and I have a deadline to meet.

We run a mechanical screening test in which the outcome can be classified as either a success, or failure. We have an existing supplier who performs well in this test, and we wish to use their level of performance as a baseline requirement as we are pleased with its field performance, and do not wish to release a product that is significantly different from it.

The tests are very expensive to run. While more tests can be run, this should be minimized to whatever extent possible. The outcomes of each test are assumed independent for a given supplier.

Using data generated in the past, I know that in 6 trials we saw no failures in this test for the current/good supplier. However it is not to be expected if the test was repeated an infinite number of times, that you would still have 0 failures.

So when we qualify new suppliers by running this test 6 times, I need to be able to quantify the probability that the supplier is worse then the current/good supplier.

For application to the equation of the Binomial distribution, if I have observed 0 failures in 6 samples with the current/good supplier, what can I state as my probability of success.

Is it: (1-0.5^6) ?

Nothing in statistics is certain, so it can't possibly be 100% with such a small sample size right?