Yeah, see, you can't do that. You're trying to change your model from inside your model; you're basically saying, "If the probability of success is 97%, and this launch was a success, what is the probability the next launch is a success?" The answer has to be 97%, because inside your model, the two launches are independent events.

You have to go meta: Instead of asking "What are the chances the next launch will succeed?" you ask, "What are the chances that my model, which predicted this launch would succeed with a probability of 97%, is correct?" To do that, you need a prior probability that the model is right, and you need other possible models. More complicated to do with this than the mammogram example, because you either have cancer or you don't, but in this example, the probability of success can be anywhere between 0 and 100%.

Imagine that your "component test and guesswork" is launching shuttle after shuttle, and seeing how many blow up. You could get the 1% figure by

• launching 100 shuttles, and observing that only one blew up
• launching 100.000 shuttles, and observing that 1000 of them blew up.

Even though both could be described as "1% likely to fail", it's clear that you have much more confidence in that figure in the second scenario; observing one extra successful launch will shift your confidence around more in the first scenario than in the second.

As others said, you should have a probability distribution over the frequency of failure (a Beta distribution I believe), that should peak near 1%, but the peak will be much sharper in the second scenario than in the first.

Good for you for actually trying to apply the Bayesian gospel to realistic problems, instead of taking it on faith!

Not sure if this is a useful example, though, as the posterior probability of a successful launch depends on a score of other tidbits of information obtained from the launch, which are bound to overwhelm the fact of successful launch itself, except in the eyes of the management, who quickly reduce the announced failure rate to one in 100,000 or some other acceptable number after only a dozen of successes, regardless of the warning signs.

In fact, the question, as stated, is meaningless, since the events S and F are mutually exclusive, and to calculate a conditional probability implicitly embedded in the Bayes expression you need compatible events. One would instead define a sample space in which S and F live (e.g. odds of an O-ring failure and its effect on the launch success).

The confidence gained from a single successful launch is no better than the confidence of seeing another head given two heads in two successive tosses of a known fair coin. Until and unless you see a really unlikely event, you should not update your priors based on the outcomes, but only based on your underlying models of the event in question and whatever useful data you can glean from the coin trajectory.

That said, you can definitely apply Bayes to discriminate between competing models of, say, isolation foam debris striking the shuttle, based on the empirical data from a given launch, which will, in turn, affect the estimate of success for the next launch.

Hmm, that ended up being wordier than I expected, but hopefully I haven't told many lies.

I think the error lies in this sentence:

"Presumably your chances of success this time are not affected by the next one being a failure."

I assume you think this is true because there's no causal relationship where the next shuttle launch can affect this one, but their successes can still be correlated, which your probability estimate isn't taking into account.

If you want to update meaningfully, you need to have an alternative hypothesis in mind. (Remember, evidence can only favor one hypothesis over another (if anything); evidence is never "for" or "against" any one theory at a time.) Perhaps the engineers believe that there is a 4% chance that any given shuttle launch will fail (H1), but you estimate a 25% chance that they're wrong and the shuttles are actually foolproof (H2). Then you estimate the probability that the first shuttle launch will fail (F) as

P(F) = P(F|H1) P(H1) + P(F|H2) P(H2) = (4%)(75%) + (0%)(25%) = 3%.

But the shuttle launch goes off ok, so now you update your opinion of the two hypotheses with Bayes' rule:

P(H1|~F) = P(~F|H1) P(H1) / P(~F) = (100% - 4%) (75%) / (100% - 3%) ≈ 74.2%.

Then your estimate that the next shuttle will fail (F') becomes:

P(F' | ~F) = P(F' | H1, ~F) P(H1 | ~F) + P(F' | H2, ~F) P(H2 | ~F)

= (4%) (74.2%) + (0%) (100% - 74.2%) ≈ 2.97%.

So the one successful shuttle launch does, in this case, lower your expectation of a failure next time. As the shuttles keep succeeding, you become gradually more and more sure that the shuttles are foolproof and the engineers are wrong. But if the launch ever does fail, you will instantly believe the engineers and assign no credence to the claim that the shuttles never fail. (Try the math to see how that works.)

You're finding a probability based on a group of independent events, so I'd suggest using beta distribution. The Wikipedia page gives how to find the mean and variance given alpha and beta. Decide what you think they are (you already effectively gave that the mean was 3%, so you just need the variance), and solve for alpha and beta. If you observe a success, increment beta, and recalculate the mean.

Also, you seem to be saying 1% failure at the beginning, and 3% later on. In addition, both of these would be much too high to risk a launch.

A few people have said things that are similar while I was typing this up, but hopefully this still helps: I think the problem is that you're implicitly assigning a probability of 0 to anything other than that one rate. In usual Bayesian analysis, you could imagine the launches as being analogous to a "biased coin." Often success/failure scenarios are modeled as binomial, with a beta distribution describing our degrees of belief for what the "coin" will do. But for simplicity's sake, let's suppose we know that either 1% or 10% of the launches will fail, and we have no further information on what will happen, so we assign a probability of 0.5 to both rates of failure. Because either one or the other has to be the case (by the unrealistic setup of this problem), the unconditional probability of a failure is P(F) = 0.5(0.01) + 0.5(0.10) = 0.055

Now, we see a successful launch. Intuitively, this is more likely if the rate of failure is lower, so this should favor, for the rate R, the hypothesis R = 0.01 and decrease the probability that R=0.1:

P(R = 0.01|S) = P(S|R =0.01)P(R = 0.01)/P(S) = 0.99(0.5)/0.945 = 0.524 And since it has to be one or the other, P(R = 0.1|S) = 1 - 0.524 = 0.476

Other people have already said most of what there is to be said, but there is also this:

Presumably your chances of success this time are not affected by the next one being a failure, so P(1S|FNT) is just P(S) = 0.97.

Not causally affected, but possibly correlated. Or anticorrelated. At any rate, the fact that the future does not causally affect the present does not establish the probabilistic independence of 1S and FNT.

Other people have already answered your question better than I can, but I wanted to let you know that at first I thought this post would be about applying Bayesian methods to working out.