2009 H2 Mathematics Paper 2 Question 7

Probability

Answers

{0.035}

This means that when the company buys a larger percentage of its components from

{A,}

the probability that a randomly chosen component that is faulty was supplied by

{A}

increases

\begin{align*} & \textrm{P}\left(\textrm{faulty}\right) \\ &= \textrm{P}\left(A, \textrm{ faulty}\right)+\textrm{P}\left(B, \textrm{ faulty}\right) \\ &= 0.25 \left(0.05\right) + 0.75\left(0.03\right) \\ &= 0.035 \; \blacksquare \end{align*}

\begin{align*} & f(p) \\ &=\textrm{P}\left(A \mid \textrm{faulty}\right) \\ &=\frac{\textrm{P}\left(A \cap \textrm{faulty}\right)}{\textrm{P}\left(\textrm{faulty}\right)} \\ &= \frac{\left(\frac{p}{100}\right) 0.05}{\left(\frac{p}{100}\right) 0.05+\left(\frac{100-p}{100}\right) 0.03} \\ &= \frac{0.05p}{0.05p + 3 - 0.03p} \\ &= \frac{0.05p}{0.02p+3} \; \blacksquare \end{align*}

\begin{align*} & f'(p) \\ &= \frac{0.05(0.02p+3)-0.02(0.05p)}{(0.02p+3)^2} \\ &= \frac{0.15}{(0.02p+3)^2} \\ \end{align*}

For

{0 \leq p \leq 100,}

{f'(p) > 0}

{f}

is an increasing function

{\blacksquare}

This means that when the company buys a larger percentage of its components from

{A,}

the probability that a randomly chosen component that is faulty was supplied by

{A}

increases

{ \blacksquare}