A precision industrial laser's cooling system maintains a temperature deviation TTT, measured in millikelvins (mK) from a reference point. The random variable T T\,T follows a continuous distribution with cumulative distribution function F(t)F(t)F(t). The graph of F(t)F(t)F(t) consists of a single straight line segment from the point (−5,0)(-5, 0)(−5,0) to the point (15,1)(15, 1)(15,1). For t<−5t < -5t<−5, F(t)=0F(t) = 0F(t)=0, and for t>15t > 15t>15, F(t)=1F(t) = 1F(t)=1.
Specify fully the probability density function f(t)f(t)f(t) of TTT.
Write down the value of E(T)E(T)E(T).
Determine the value of k k\,k such that P(2.5≤T≤k)=0.35P(2.5 \le T \le k) = 0.35P(2.5≤T≤k)=0.35.
One operating hour is selected at random.
Calculate the probability that the temperature deviation is between 6 mK and 10 mK.
Given that the temperature deviation was between 6 mK and 10 mK, calculate the probability that it was greater than 9.1 mK.
A random sample of 40 operating hours is taken.
Calculate the probability that for at most 5 of these hours the temperature deviation is between 6 mK and 10 mK.
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.