# How does the posterior of a dirac prior look like?

Edit for the Moderators: Should this question migrate to stats.stackexchange?

I have a very basic question concerning updating from a prior to a posterior in bayesian statistics.

Setting:

• I enter in a casino and I have a degenerate prior over an american roulette wheel, say $\delta_{\{0\}}$.

• The outcome of the first roll I witness is not $0$.

[Notation: $\delta_{\{0\}}$ denotes the dirac measure associated with the belief that the roulette is unfair and completely bias towards $0$]

Questions:

1. How do I update my degenerate prior?
2. Is my posterior the uniform over $\{ 0, 1 , \dots , 36 \}$?.
3. What kind of prior is it?
To what kind of posterior does it lead?
[I assume there are some technical names for these things, that I am unaware of]

Of course the questions come from the fact that it is not possible to update anything with such a prior, if the hard evidence I can obtain does not support it.

Any feedback is most welcome.
1. Using Bayes' rule, for each value $x$ you have $$P(x | data) \propto P(data | x) prior(x)$$ Therefore, $P(x|data)$ is zero if $x \neq 0$ and so $P(0|data) = 1$, so your posterior is the same as the prior.