Specifically, Laplace's method assumes the uniform prior. If you do the same thing and apply the Beta Distribution to this problem, you can answer the question

Specifically, Laplace's method assumes the uniform prior. If you do the same thing and apply the Beta Distribution to this problem, you can answer the question "what is the probability that the coin lands on heads more than X% of the time. The answer, assuming a uniform prior and 9 heads, is that there's a 65% chance, for example, that the coin is biased more than 90% in favor of heads. If on the other hand you choose a different prior, for example "the coin may have two heads, and it's difficult to manufacture a biased coin with two different faces that still favors one side more than 90% of the time", then you might instead reach the conclusion that there's a 65% chance that the coin has two heads. But we don't usually deal with weird priors like that in intro stats. If, as GP suggests, you know that the coin is somehow biased, you should seed your beta distribution with a non-uniform prior. That is Beta(.5, .5)[0] or something instead of Beta(1, 1). Then you'd be 84% sure, instead of 65% sure that the coin was biased at least 90% in favor of heads. [0]: https://homepage.divms.uiowa.edu/~mbognar/applets/beta.html https://homepage.divms.uiowa.edu/~mbognar/applets/beta.html Play around with this, in the Beta distribution, a=success, b=failures, X is the mean of the underlying dataset, and P(X>x) is the propbality that the mean of the underlying dataset is greater than the provided value. Look at - Beta(1,1): the uniform prior - Beta(10, 1): Laplace's method - Beta(.5, .5): A prior where you believe the coin is weighted, but aren't sure how much. - Beta(.1, .1): A prior where you're very confident the coin is weighted, but unsure in which direction - Beta(9.5, .5): The coin after 9 flips - Beta(9.1, .1): The coin after 9 flips when you're really confident it's weighted - Beta(8.1, 1.1): The coin after 9 flips when you're really confident it's weighted, but both sides aren't in fact the same. Those will give you some intuition about the distribution itself.