2007 H2 Mathematics Paper 2 Question 10

Probability

Answers

(i)

164{\frac{1}{64}}

(ii)

21256{\frac{21}{256}}

(iii)

1317{\frac{13}{17}}

Full solutions

(i)

P(all three successful)=18×14×12=164  ■\begin{align*} & \textrm{P}\left(\textrm{all three successful}\right) \\ &= \frac{1}{8} \times \frac{1}{4} \times \frac{1}{2} \\ &= \frac{1}{64} \; \blacksquare \end{align*}

(ii)

P(hit, hit, miss)=18×14×12=164\begin{align*} & \textrm{P}\left(\textrm{hit, hit, miss}\right) \\ &= \frac{1}{8} \times \frac{1}{4} \times \frac{1}{2} \\ &= \frac{1}{64} \end{align*}
P(hit, miss, hit)=18×34×14=3128\begin{align*} & \textrm{P}\left(\textrm{hit, miss, hit}\right) \\ &= \frac{1}{8} \times \frac{3}{4} \times \frac{1}{4} \\ &= \frac{3}{128} \end{align*}
P(miss, hit, hit)=78×18×14=7256\begin{align*} & \textrm{P}\left(\textrm{miss, hit, hit}\right) \\ &= \frac{7}{8} \times \frac{1}{8} \times \frac{1}{4} \\ &= \frac{7}{256} \end{align*}
P(at least two successful)=164+164+3128+7256=21256  ■\begin{align*} & \textrm{P}\left(\textrm{at least two successful}\right) \\ &= \frac{1}{64} + \frac{1}{64} + \frac{3}{128} + \frac{7}{256} \\ &= \frac{21}{256} \; \blacksquare \end{align*}

(iii)

P(third throw successful∣exactly two throws successful)=P(third throw successful∩exactly two throws successful)P(exactly two throws successful)=P(hit, miss, hit)+P(miss, hit, hit)P(hit, miss, hit)+P(miss, hit, hit)+P(hit, hit, miss)=3128+7256164+3128+7256=1317  ■\begin{align*} &\textrm{P}\left(\textrm{third throw successful} \mid \textrm{exactly two throws successful}\right) \\ &=\frac{\textrm{P}\left(\textrm{third throw successful} \cap \textrm{exactly two throws successful}\right)}{\textrm{P}\left(\textrm{exactly two throws successful}\right)} \\ &= \frac{\textrm{P}\left(\textrm{hit, miss, hit}\right)+\textrm{P}\left(\textrm{miss, hit, hit}\right)}{\textrm{P}\left(\textrm{hit, miss, hit}\right)+\textrm{P}\left(\textrm{miss, hit, hit}\right)+\textrm{P}\left(\textrm{hit, hit, miss}\right)} \\ &= \frac{\frac{3}{128}+\frac{7}{256}}{\frac{1}{64}+\frac{3}{128}+\frac{7}{256}} \\ &= \frac{13}{17} \; \blacksquare \end{align*}