Bayesian updating of a constant probability using one data point A reformulation of a question that came up in a model:
Imagine a toy store that sells $K$ toys, where our prior is that each toy has equal probability $1/K$ of being purchased by a customer. Then you have a customer come in and buy a teddy bear. Can we use Bayesian updating to find a new probability estimate of a customer buying a teddy bear?

This is how I read your question: when a customer comes to a store, they always buy one, and only one, of $K$ items. There’s an infinite supply of the items (i.e. they are sampled with replacement). With no other prior knowledge, you assume a prior probability for picking any of the items to be $1/K$. A customer comes and buys $i$-th item, you want to use this information to update the prior.
In such a case, use Dirichlet-multinomial model. Your prior is an uniform Dirichlet distribution with parameters $\alpha_1=\alpha_2=\dots= \alpha_K=1$, hence the individual probabilities are on average equal to $1/K$. When you observe a purchase for the $i$-th category, you update the prior to get the posterior $E[p_i]=\frac{2}{K + 1}$ and $\frac{1}{K+1}$ for other categories.
