2024 H2 Mathematics Paper 2 Question 7

Probability

Answers

(a)

$\frac{2}{n} .$

$\frac{2}{n} .$

$n \in ℤ, n \geq 37.$

$x = \frac{1}{14} .$

Full Solutions

(a)

(i)

\begin{aligned} P (A finds the prize) \\ = \frac{1}{n} + \frac{n - 1}{n} \frac{1}{n - 1} \\ = \frac{2}{n} ∎ \end{aligned}

(ii)

\begin{aligned} P (B finds the prize) \\ = (1 - \frac{2}{n}) \frac{1}{n - 2} + (1 - \frac{2}{n}) \frac{n - 3}{n - 2} \frac{1}{n - 3} \\ = \frac{n - 2}{n} \frac{1}{n - 2} + \frac{n - 2}{n} \frac{1}{n - 2} \\ = \frac{2}{n} ∎ \end{aligned}

(iii)

\begin{aligned} P (C finds the prize) & > 8 P (A or B finds the prize) \\ 1 - \frac{2}{n} - \frac{2}{n} & > 8 (\frac{2}{n} + \frac{2}{n}) \\ 1 - \frac{4}{n} & > \frac{32}{n} \\ \frac{32}{n} + \frac{4}{n} & < 1 \\ \frac{36}{n} & < 1 \\ n & > 36 \end{aligned}

Since $n$ is an integer, the range of values of $n$ is:

n \in ℤ, n \geq 37 ∎

(iv)

\begin{aligned} P (Q \cap R^{'}) & = P (Q) \\ = x + 0.1 \end{aligned}

venn-diagram

\begin{aligned} P ((Q \cup R)^{'}) & = P (Q) + 2 P (R) \\ 1 - 0.1 - x - (x + 0.1) & = (x + 0.1) + 2 (x + x + 0.1) \\ 0.8 - 2 x & = 5 x + 0.3 \\ 7 x & = 0.5 \\ x & = \frac{1}{14} ∎ \end{aligned}