# How can I calculate Posterior Distribution, analytically with given information?

My apology for stating my question with an image. The following image shows that the posterior distribution is as follows with the given information:

I wonder how the posterior has been calculated, analytically.

• You should explain why you cannot compute $p(\theta|x)$ from $p(\theta)$ and $p(x|\theta)$. Commented May 13, 2022 at 6:57
• I am new to Bayesain Statistics and trying to understand how the above question is derived from the given information. Commented May 13, 2022 at 7:07
• You should at least know how to move from the pair prior x likelihood to the posterior Commented May 13, 2022 at 7:15
• I gave it a go and solved the question. When I uploaded the question, I didn't have a clue how to calculate it but now I can. Thank you for your comment. Commented May 13, 2022 at 7:17

For $$x=1$$
$$\dfrac{p(\theta)p(x=1 | \theta)}{p(\theta_1)p(x=1 | \theta) + p(\theta_2)p(x=1 | \theta_2)} = \dfrac{p(0)p(x=1 | 0)}{p(0)p(x=1|0)+p(1)p(x=1|1)} = \dfrac{0.1665}{0.417} = 0.399280576 \approx 0.4$$
So, when $$x=0$$, the probability of $$x=0$$ is $$0.6$$
• The use of slightly incorrect decimal approximations, written out in excessive precision, makes this calculation somewhat confusing: the answer $3/5$ is exact, not approximate.