# Probability of transmitting a signal through a network of transmitters.

We have a network of four transmitters $$A$$, $$B$$, $$C$$ and $$D$$. What is the probability of transmitting a signal through the network if all transmitters work independently and the probabilities of transmitting a signal by each transmitter are $$0.7$$, $$0.8$$, $$0.9$$ adn $$0.6$$, respectively. My take is the following. The probability of transmitting a signal through $$C$$ and then $$D$$ is $$P(C \cap D) = 0.9 \cdot 0.6 = 0.54$$. The event "a signal transmitted through $$A$$ or $$B$$" consists of three disjoint events: $$\overline A \cap B$$, $$A \cap \overline B$$ and $$A \cap B$$. We have $$P(A \cup B) = P(\overline A \cap B) + P(A \cap \overline B) + P(A \cap B) = 0.3 \cdot 0.8 + 0.7 \cdot 0.2 + 0.7 \cdot 0.8 = 0.94$$.

So the probability of transmitting a signal through the entire network is: $$P(A \cup B) \cdot P(C \cap D) = 0.94 \cdot 0.54 = 0.5076$$. Is that correct?

Another calculation could be $$P\left(\overline{(\overline A \cap \overline B)} \cap C \cap D\right)=(1-(1-0.7)\times (1-0.8))\times 0.9\times0.6 = 0.5076$$