A sender (S) transmits a signal, which can be one of the two kinds: $H$ and $L$ with probabilities 0.1 and 0.9 respectively, to a receiver (R). In the graph below, the weight of edge $(u, v)$ is the probability of receiving $v$ when $u$ is transmitted, where $u, v \in \{H, L\}$. For example, the probability that the received signal is $L$ given the transmitted signal was $H$, is 0.7. If the received signal is $H$, the probability that the transmitted signal was $H$ (rounded to 2 decimal places) is ___________.

Let $T$ denote the transmitted signal and $R$ denote the received signal.

Given prior probabilities:

• $P(T = H) = 0.1$

• $P(T = L) = 0.9$

Conditional probabilities from the transition graph:

• $P(R = H \mid T = H) = 0.3$

• $P(R = L \mid T = H) = 0.7$

• $P(R = H \mid T = L) = 0.8$

• $P(R = L \mid T = L) = 0.2$

By the Law of Total Probability, the total probability of receiving $H$ is:

Using Bayes' Theorem, the posterior probability $P(T = H \mid R = H)$ is:

Rounded to 2 decimal places, the answer is 0.04.