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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.

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Solution

## $P\left(\text{A grade in Maths}\right)=P\left(A\right)=0.2\phantom{\rule{0ex}{0ex}}P\left(\text{A grade in Physics}\right)=P\left(B\right)=0.3\phantom{\rule{0ex}{0ex}}P\left(\text{A grade in Chemistry}\right)=P\left(C\right)=0.5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{i}\right)P\left(\text{grade A in all subjects}\right)=P\left(A\right)×P\left(B\right)×P\left(C\right)\phantom{\rule{0ex}{0ex}}=0.2×0.3×0.5\phantom{\rule{0ex}{0ex}}=0.03\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right)P\left(\text{grade A in no subject}\right)=P\left(\overline{A}\right)×P\left(\overline{B}\right)×P\left(\overline{C}\right)\phantom{\rule{0ex}{0ex}}=\left(1-0.2\right)×\left(1-0.3\right)×\left(1-0.5\right)\phantom{\rule{0ex}{0ex}}=0.8×0.7×0.5\phantom{\rule{0ex}{0ex}}=0.28\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{iii}\right)P\left(\text{grade A in two subjects}\right)=P\left(\text{not grade A in Maths}\right)+P\left(\text{not grade A in Physics}\right)+P\left(\text{not grade A in Chemistry}\right)\phantom{\rule{0ex}{0ex}}=P\left(\overline{)A}\right)×P\left(B\right)×P\left(C\right)+P\left(A\right)×P\left(\overline{)B}\right)×P\left(C\right)+P\left(A\right)×P\left(B\right)×P\left(\overline{)C}\right)\phantom{\rule{0ex}{0ex}}=\left(1-0.2\right)×0.3×0.5+0.2×\left(1-0.3\right)×0.5+0.2×0.3×\left(1-0.5\right)\phantom{\rule{0ex}{0ex}}=0.8×0.3×0.5+0.2×0.7×0.5+0.2×0.3×0.5\phantom{\rule{0ex}{0ex}}=0.12+0.07+0.03\phantom{\rule{0ex}{0ex}}=0.22$

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