Example: Probability of rain tmr

A: My leg itches when it rains and its kinda itchy $ \rightarrow p = .8 $

A: Done complex calculations and have seens 10,451 days like tmr $ \rightarrow p = .8 $

Which estimate is better? We don’t have a measure for confidence?

Belief in $ p $ After 9 Heads in 10 Flips?

Flip a coin 10 times, observed 9 H and 1 T, $ p \approx \frac{9}{10} $?

• A very rough estimate based off 10 coin flips
• the ‘point value’ $ \frac{9}{10} $ does not have the abilityy to articulate how uncertain it is.

$ \Rightarrow $ Allows probability of heads to be a r.v itself.

Let $ X $ be the probability of heads

$$ \begin{align*} f(X &= x \mid H = 9, T = 1), \quad \quad 0 < x < 1 \\ &= \frac{P(H = 9, T = 1 \mid X = x) f(X = x)}{\int_0^1 f(H = 9, T = 1 \mid X = x) f(X = x)dx} \end{align*} $$