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JGWeissman comments on Bayesian exercise - Less Wrong Discussion
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Your problem is that you are effectively assigning probability 1 to the proposition that 1% of launches will fail. Instead, you should have a probability distribution over the fraction of launches that fail. When you observe a launch or a failure, update that probability distribution using Bayes' law, resulting in higher probabilities for lower frequencies after a success.
It occurs to me that I don't really know how to mathematically handle a probability distribution. How much calculus, if any, is required for this?
You need calculus if you're going to try to estimate any continuous quantities, but you can often avoid this by making the variable discrete. Instead of saying "the proportion is a number [0,1]" you say "the proportion is either 0, .25, .5, .75 or 1". This approximates the continuous version and can be done without any calculus.
To fully interpret a probability distribution you need to use integrals. For example, if I have a probability distribution over the number of heads in 50 coinflips and I want to know the probability that the observed value is going to fall within a certain interval, I have to take the integral of that part of the distribution. You can definitely understand what a probability distribution is without calculus, but you're going to have a hard time actually doing the math.
Edit: It occurs to me that statistical software could do most of the number-crunching for you, which would definitely make things easier.
For probability distributions on continuous quantities (such as the proportion of launches that fail), you need to know how to do derivatives and integrals.