The continuous random variable WWW is used to model the endurance of a high-performance industrial drill bit, where www represents the number of thousands of cycles completed before the bit requires replacement.
The probability that a drill bit lasts for more than www thousand cycles is given by
P(W>w)=16(w+4)2,w≥0 P(W > w) = \frac{16}{(w + 4)^2}, \quad w \ge 0 P(W>w)=(w+4)216,w≥0Show that the cumulative distribution function (CDF) of WWW is given by
F(w)={1−16(w+4)2w≥00otherwise F(w) = \begin{cases} 1 - \frac{16}{(w+4)^2} & w \ge 0 \\ 0 & \text{otherwise} \end{cases} F(w)={1−(w+4)2160w≥0otherwiseDetermine the probability that a randomly selected drill bit will fail within the first 4000 cycles.
Given that a drill bit has already completed 4000 cycles, find the probability that it will last for at least 8000 more cycles.
Calculate the number of cycles after which only 6.25%6.25\%6.25% of these drill bits 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.