Suppose I want to learn the value of $$\omega\in\{0,1\}$$. I observe a sequence of iid signals $$(s_n)_{n\ge1}$$ with $$\Pr(s_n=0\,\vert\,\omega=0)=1-\alpha$$ and $$\Pr(s_n=1\,\vert\,\omega=1)=1-\beta,$$ where $$\alpha$$ and $$\beta$$ are false positive and false negative rates. I let $$\pi_n$$ denote my belief that $$\omega=1$$ after observing $$n$$ signals, and update this belief sequentially via Bayes’ formula: $$\pi_{n}(s)=\frac{\Pr(s_n=s\,\vert\,\omega=1)\pi_{n-1}}{\Pr(s_n=s)}.$$ In particular, if I observe $$s_n=0$$ then I update my belief to $$\pi_n(0)=\frac{\beta\pi_{n-1}}{\beta\pi_{n-1}+(1-\alpha)(1-\pi_{n-1})},$$ whereas if I observe $$s_n=1$$ then I update my belief to $$\pi_n(1)=\frac{(1-\beta)\pi_{n-1}}{(1-\beta)\pi_{n-1}+\alpha(1-\pi_{n-1})}.$$

The chart below shows how my belief $$\pi_n$$ changes with $$n$$. Each path in the chart corresponds to the sequence of beliefs $$(\pi_0,\pi_1,\ldots,\pi_{100})$$ obtained by updating my initial belief $$\pi_0=0.5$$ in response to a signal sequence $$(s_1,s_2,\ldots,s_{100})$$. I simulate 10 such sequences, fixing $$\omega=1$$ and $$\alpha=0.4$$ but varying $$\beta\in\{0.2,0.4,0.6,0.8\}$$. If $$\beta\not=0.6$$ then my belief converges to $$\pi_n=1$$ as $$n$$ grows. However, if $$\beta=0.6$$ then $$\pi_n=\pi_0$$ for each $$n$$; that is, I never update my beliefs regardless of the signals I observe. This is because if $$\alpha+\beta=1$$ then $$\Pr(s_n=s\cap\omega=1)=\Pr(s_n=s)$$ for each $$s\in\{0,1\}$$, and so signals are uninformative because they are independent of $$\omega$$.

The chart below plots the mean of my beliefs $$\pi_n$$ across 1,000 realizations of the signals simulated above. Again, I fix $$\omega=1$$ and the false positive rate $$\alpha=0.4$$ but vary the false negative rate $$\beta\in\{0.2,0.4,0.6,0.8\}$$. Higher values of $$\beta$$ are not always worse: my belief converges to the truth faster when $$\beta=0.8$$ than when $$\beta=0.4$$. Intuitively, if I know the false negative rate is close to 100% then observing a signal $$s_n=0$$ gives me strong evidence that $$\omega=1$$. 