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