Let A, B, C, D are the events of solving a problem.

Given: $$\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{C})=\frac{1}{4}, \mathrm{P}(\mathrm{D})=\frac{1}{8}$$

Also given A, B, C, D are independent events.

Let P is the probability of solved by at least one person.

$$\Rightarrow \mathrm{P}=1-$$ (Probability that none of the person can solve the problem)

$$\begin{aligned} & \Rightarrow \mathrm{P}=1-\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}} \cap \overline{\mathrm{C}} \cap \overline{\mathrm{D}}) \\ & \Rightarrow \mathrm{P}=1-\mathrm{P}(\overline{\mathrm{A}}) \cdot \mathrm{P}(\overline{\mathrm{B}}) \cdot \mathrm{P}(\overline{\mathrm{C}}) \cdot \mathrm{P}(\overline{\mathrm{D}}) \\ & \Rightarrow \mathrm{P}=1-(1-\mathrm{P}(\mathrm{A}))(1-\mathrm{P}(\mathrm{B}))(1-\mathrm{P}(\mathrm{C}))(1-\mathrm{P}(\mathrm{D})) \\ & \Rightarrow \mathrm{P}=1-\left(1-\frac{1}{2}\right)\left(1-\frac{3}{4}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{8}\right) \\ & \Rightarrow \mathrm{P}=1-\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4} \times \frac{7}{8} \\ & \Rightarrow \mathrm{P}=1-\frac{21}{256}\\ & \Rightarrow \mathrm{P}=\frac{235}{256} \end{aligned}$$

Hints:

$$\Rightarrow$$ (i) If $$A$$ and $$B$$ are independent events, then

$$\begin{aligned} & \rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \\ & \rightarrow \mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=\mathrm{P}(\overline{\mathrm{A}}) \cdot \mathrm{P}(\overline{\mathrm{B}})=(1-\mathrm{P}(\mathrm{A}))(1-\mathrm{P}(\mathrm{B})) \\ & \rightarrow \mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})=\mathrm{P}(\overline{\mathrm{A}}) \cdot \mathrm{P}(\mathrm{B})=(1-\mathrm{P}(\mathrm{A})) \mathrm{P}(\mathrm{B}) \\ & \rightarrow \mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\overline{\mathrm{B}})=\mathrm{P}(\mathrm{A}) \cdot(1-\mathrm{P}(\mathrm{B})) \end{aligned}$$

(ii) Probability of at least one person can solve the problem $$= 1-$$ Probability that none of the persons can solve that problem.