Question

A sender \( S \) transmits a signal which can be one of two kinds: \( H \) and \( L \) with probabilities 0.1 and 0.9 respectively, to a receiver \( R \). The weight of edge \( (u,v) \) is the probability of receiving \( v \) when \( u \) is transmitted. If the received signal is \( H \), the probability that the transmitted signal was \( H \) (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\). 

Hint:

Bayes' theorem is used to reverse conditional probabilities using prior probabilities.

Answer 1

Correct Answer: 0.04

Step 1: Identify given probabilities.
\[ P(H) = 0.1, P(L) = 0.9 \] From the graph:
\[ P(R=H \mid S=H) = 0.3, P(R=H \mid S=L) = 0.8 \]

Step 2: Apply Bayes' theorem.
\[ P(S=H \mid R=H) = \frac{P(R=H \mid S=H)\,P(S=H)}{P(R=H)} \]

Step 3: Compute total probability of receiving \( H \).
\[ P(R=H) = (0.3)(0.1) + (0.8)(0.9) \] \[ P(R=H) = 0.03 + 0.72 = 0.75 \]

Step 4: Compute conditional probability.
\[ P(S=H \mid R=H) = \frac{0.03}{0.75} = 0.04 \] % Final Answer

Final Answer: \[ \boxed{0.04} \]