The continuous random variable HHH is used to model the operational lifespan in months, hhh, of a specialized high-pressure seal used in deep-sea exploration probes.
The probability that a seal remains functional for more than hhh months is given by the survival function
P(H>h)=64(h+8)2,h≥0 P(H > h) = \frac{64}{(h+8)^2}, \quad h \ge 0 P(H>h)=(h+8)264,h≥0Show that the cumulative distribution function (CDF) of HHH is given by
F(h)={1−64(h+8)2h≥00otherwise F(h) = \begin{cases} 1 - \frac{64}{(h+8)^2} & h \ge 0 \\ 0 & \text{otherwise} \end{cases} F(h)={1−(h+8)2640h≥0otherwiseDetermine the probability that a randomly selected seal will fail within the first 2 months of operation.
Given that a seal has already remained functional for 2 months, find the probability that it will last for at least 30 more months.
Calculate the number of months after which only 25%25\%25% of these seals are expected to remain functional.
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.