probability interview questions of posterior probability of two unfair coins

Question

Here is a probability interview questions: two independent unfair coins with probabilities of head, $p_1,p_2\sim U(0,1).$ Then what's the probability of $p_1>p_2$ if we tossed each coin 3 times with 2 head for $p_1$ and 1 head for $p_2?$ We simply denote as:

$$P(p_1>p_2| 2H^1,1H^2).$$

I know we should use Bayes formula: $$P(p_1>p_2| 2H^1,1H^2)P(2H^1)P(1H^1) = P(p_1>p_2,2H^1, 1H^2).$$

And I know we can calculate $P(2H^1),P(1H^2)$ by $$P(2H^1) = \int^1_0p_1P(2H^1|p_1)dp_1.$$

But how to calculate the joint distribution $P(p_1>p_2,2H^1, 1H^2)?$ Am I right to calculate as: $$P(p_1>p_2,2H^1, 1H^1) = \int_{p_1,p_2\in [0,1],p_1 > p_2}P(2H^1|p_1)P(1H^2|p_2)dp_1dp_2?$$

Answer 1

Yes, the integral is correct. We can see it more explicitly, using total probability theorem from the beginning:

$$\begin{align}\mathbb P(p_1>p_2,2H^1,1H^2)&=\int_{[0,1]\times[0,1]} \mathbb P(p_1>p_2,2H^1,1H^2|p_1,p_2)\underbrace{f_{P_1,P_2}(p_1,p_2)}_{f_{P_1}(p_1)f_{P_2}(p_2)=1}dp_1dp_2\\&=\int_{[0,1]\times[0,1]}\mathbb P(p_1>p_2|p_1,p_2)\mathbb P(2H^1|p_1,p_2)\mathbb P(1H^2|p_1,p_2)dp_1dp_2\\&=\int_{[0,1]\times[0,1]} \mathbb I(p_1>p_2)\mathbb P(2H^1|p_1)\mathbb P(1H^2|p_2)dp_1dp_2\\&=\int_{p_1,p_2\in [0,1] \cap p_1>p_2}\mathbb P(2H^1|p_1)\mathbb P(1H^2|p_2)dp_1dp_2\end{align}$$