Continuous distributions

Sample questions

Original practice questions written in the style of the exam.

1. Question 14 marks

   A stick of length $20\,\text{cm}$ is broken at a point chosen at random along its length. Find the probability that the longer piece is at least three times as long as the shorter piece.

   Mark scheme

   • B1 Distance $X$ of the break from one end: $X \sim U[0, 20]$ (1)
   • M1 Longer $\ge 3 \times$ shorter means the shorter piece is at most $5\,\text{cm}$ (1)
   • M1 So $X \le 5$ or $X \ge 15$ (1)
   • A1 Probability $= \tfrac{5 + 5}{20} = 0.5$ (1)
2. Question 25 marks

   $X \sim B(80, 0.4)$. Use a suitable normal approximation to estimate $P(X \ge 36)$.

   [You may use $\Phi(0.80) = 0.7881$.]