The continuous random variable T T\,T represents the operational lifespan, in days, of a bioluminescent underwater sensor. The probability density function of T T\,T is modeled by:
f(t)={c(25−t2)0≤t≤50otherwise f(t) = \begin{cases} c(25 - t^2) & 0 \le t \le 5 \\ 0 & \text{otherwise} \end{cases} f(t)={c(25−t2)00≤t≤5otherwisewhere c c\,c is a constant.
Show that c=3250\displaystyle c = \frac{3}{250}c=2503.
Determine the cumulative distribution function F(t)F(t)F(t) for the interval 0≤t≤50 \le t \le 50≤t≤5.
Calculate the probability that a randomly selected sensor remains operational for more than 3 days.
Given that a sensor has already functioned for 3 days, determine the probability that it will last for at least 4 days in total.
Five such sensors are deployed independently. Find the probability that exactly 2 of them remain operational for more than 3 days.
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.