Question

X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets
(i) Grade A in all subjects
(ii) Grade A in no subject
(iii) Grade A in two subjects.

Answer 1

$$\begin{gathered} P\left(\text{A grade in Maths}\right)=P\left(A\right)=0.2 \\ P\left(\text{A grade in Physics}\right)=P\left(B\right)=0.3 \\ P\left(\text{A grade in Chemistry}\right)=P\left(C\right)=0.5 \\ \left(\mathrm{i}\right)P\left(\text{grade A in all subjects}\right)=P\left(A\right)\times P\left(B\right)\times P\left(C\right) \\ =0.2\times 0.3\times 0.5 \\ =0.03 \\ \left(\mathrm{ii}\right)P\left(\text{grade A in no subject}\right)=P\left(\overline{A}\right)\times P\left(\overline{B}\right)\times P\left(\overline{C}\right) \\ =\left(1-0.2\right)\times \left(1-0.3\right)\times \left(1-0.5\right) \\ =0.8\times 0.7\times 0.5 \\ =0.28 \\ \left(\mathrm{iii}\right)P\left(\text{grade A in two subjects}\right)=P\left(\text{not grade A in Maths}\right)+P\left(\text{not grade A in Physics}\right)+P\left(\text{not grade A in Chemistry}\right) \\ =P\left(\overline{A}\right)\times P\left(B\right)\times P\left(C\right)+P\left(A\right)\times P\left(\overline{B}\right)\times P\left(C\right)+P\left(A\right)\times P\left(B\right)\times P\left(\overline{C}\right) \\ =\left(1-0.2\right)\times 0.3\times 0.5+0.2\times \left(1-0.3\right)\times 0.5+0.2\times 0.3\times \left(1-0.5\right) \\ =0.8\times 0.3\times 0.5+0.2\times 0.7\times 0.5+0.2\times 0.3\times 0.5 \\ =0.12+0.07+0.03 \\ =0.22 \end{gathered}$$