Question

a circular target of radius 11 cm consists of an inner circleof radius 5 cm and three concentric circles of radii 7cm, 9cm and 10 cm dividing the target into four regions. If a shot hit the target find the probability of hitting each region. If you shoot target 121 times, what is your expectation? How can you improve your performance?

Answer 1

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{circle}\mathrm{with}\mathrm{r}=11\mathrm{cm} \\ \frac{22}{7}\times (11{)}^2 \\ \mathrm{Area}\mathrm{of}\mathrm{region}1,\frac{22}{7}({11}^2-{10}^2)=66{\mathrm{cm}}^2 \\ \mathrm{Area}\mathrm{of}\mathrm{region}2,\frac{22}{7}({10}^2-9^2)=\frac{22}{7}\times 19{\mathrm{cm}}^2 \\ \mathrm{Area}\mathrm{of}\mathrm{region}3,\frac{22}{7}(9^2-7^2)=\frac{22}{7}\times 32{\mathrm{cm}}^2 \\ \mathrm{Area}\mathrm{of}\mathrm{region}4,\frac{22}{7}\times 7^2=154{\mathrm{cm}}^2 \\ \mathrm{Probability}\mathrm{of}\mathrm{hitting}\mathrm{region}1,\frac{66\times 7}{22\times {11}^2}=\frac{21}{121} \\ \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}2,\frac{{\displaystyle \frac{22}{7}\times 19}}{{\displaystyle \frac{22}{7}\times {11}^2}}=\frac{19}{121} \\ \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}3,\frac{{\displaystyle \frac{22}{7}\times 32}}{{\displaystyle \frac{22}{7}\times {11}^2}}=\frac{32}{121} \\ \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}4,\frac{154}{{\displaystyle \frac{22}{7}\times {11}^2}}=\frac{49}{121} \end{gathered}$$

Expectation of hitting in the following regions:
Region 1: 21
Region 2:19
Region 3: 32
Region 4: 49

To improve his performance he needs to practice more.