A communications engineer monitors the stability of a satellite link. Let event UUU represent a failure in the uplink and event DDD represent a failure in the downlink during a specific transmission window. The following probabilities are recorded:
P(U)=0.25,P(U∩D)=0.05andP(U′∩D′)=0.72 P(U) = 0.25, \quad P(U \cap D) = 0.05 \quad \text{and} \quad P(U' \cap D') = 0.72 P(U)=0.25,P(U∩D)=0.05andP(U′∩D′)=0.72Find the probability that during a transmission window:
At least one type of failure (uplink or downlink) occurs.
A downlink failure occurs.
Given that an uplink failure has occurred, find the probability that a downlink failure also occurs.
The engineer suspects that the occurrence of an uplink failure might influence the likelihood of a downlink failure. Determine whether or not events UUU and DDD are statistically independent.
Comment on the engineer's suspicion in light of your answer to part (d).