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).
200 exam-style questions on OCR (MEI) A Level Maths 2.3 Probability, covering 2.3.1 Calculate the probability of an event, 2.3.2 Complementary events, 2.3.3 Expected frequency of an event, 2.3.4 Diagrams to calculate probabilities, 2.3.5 Mutually exclusive and independent events, 2.3.6 Add probabilities for mutually exclusive events, 2.3.7 Multiply probabilities for independent events, 2.3.8 Mutually exclusive and independent events (notation) (A-level only), 2.3.9 Venn diagrams for probabilities (A-level only), 2.3.10 Conditional probabilities (A-level only), and 2.3.11 Independence via conditional probability (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.