The processing time TTT (in microseconds) of a cryptographic hashing algorithm is modeled by a continuous random variable with probability density function:
f(t)={t200≤t≤k10−t30k<t≤100otherwise f(t) = \begin{cases} \frac{t}{20} & 0 \le t \le k \\ \frac{10-t}{30} & k < t \le 10 \\ 0 & \text{otherwise} \end{cases} f(t)=⎩⎨⎧20t3010−t00≤t≤kk<t≤10otherwiseBy considering the total area under the probability density function, show that k k\,k satisfies the equation k2−8k+16=0k^2 - 8k + 16 = 0k2−8k+16=0 and hence find the value of kkk.
Determine the mode of TTT.
Find P(T≤k2∣T≤k)\displaystyle P\left( T \le \frac{k}{2} \mid T \le k \right)P(T≤2k∣T≤k).
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.