The continuous random variable TTT is used to model the lifespan in hours, ttt, of a specific type of chemical reactant after it is activated.
The probability that the reactant lasts for more than ttt hours is given by
P(T>t)=100(t+10)2,t≥0 P(T > t) = \frac{100}{(t+10)^2}, \quad t \ge 0 P(T>t)=(t+10)2100,t≥0Show that the cumulative distribution function (CDF) of TTT is given by
F(t)={1−100(t+10)2t≥00otherwise F(t) = \begin{cases} 1 - \frac{100}{(t+10)^2} & t \ge 0 \\ 0 & \text{otherwise} \end{cases} F(t)={1−(t+10)21000t≥0otherwiseFind the probability that a randomly selected reactant will expire within 5 hours of activation.
Given that a reactant has already lasted for 5 hours, find the probability that it will last for at least 15 more hours.
Calculate the number of hours after which only 25%25\%25% of these reactants are expected to remain active.
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.