Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. Suppose that the probability of producing a defective product follows the Beta distribution
$$\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(x) = \frac{1}{\mathrm{B}(\mathrm{a},\mathrm{b})} x^{\mathrm{a}-1}(1-x)^{\mathrm{b}-1} \quad (0 \leq x \leq 1),$$
for real numbers $\mathrm{a}(>1)$ and $\mathrm{b}(>1)$. Note that the Beta function $\mathrm{B}(\mathrm{a},\mathrm{b})$ is defined as
$$\mathrm{B}(\mathrm{a},\mathrm{b}) = \int_0^1 t^{\mathrm{a}-1}(1-t)^{\mathrm{b}-1} \mathrm{d}t$$
In the Bayesian estimation, the parameter $\theta$ (in this case, $\phi$) that determines the probability is treated as the random variable and we assume that its distribution is described by $\pi(\theta)$. We calculate $\pi(\theta \mid A)$ by
$$\pi(\theta \mid A) = \frac{\pi(\theta) P(A \mid \theta)}{P(A)}$$
where $\pi(\theta \mid A)$ is the posterior probability, $P(A \mid \theta)$ is the conditional occurrence probability that the event $A$ is observed under $\theta$, and $\pi(\theta)$ is the prior probability.
We assume that $\phi$, the probability of producing a defective product, follows the prior probability $\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(\phi)$. Let $Q(\boldsymbol{v} \mid \phi)$ be the conditional occurrence probability of $\boldsymbol{v}$ under $\phi$ and $Q_{\mathrm{a},\mathrm{b}}(\boldsymbol{v})$ be the occurrence probability of $\boldsymbol{v}$. Obtain the posterior probability after $\boldsymbol{v}$ occurs.