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The probabilities that a student passes in mathematics, physics and chemistry are $\mathrm{ m, p}$ and $\mathrm{ c}$ respectively. Of these subjects, a student has a $\mathrm{ 75 \%}$ chance of passing in at least one, a $\mathrm{50 \%}$ chance of passing in at least two, and a $\mathrm{40 \%}$ chance of passing in exactly two subjects. Which of the following relations are true?
Option: 1
$\mathrm{p+m+c=\frac{19}{20} }$

Option: 2
$\mathrm{p+m+c=\frac{27}{20}}$

Option: 3
$\mathrm{p m c=\frac{4}{10} }$

Option: 4
$\mathrm{p m c=\frac{1}{4}}$

Answers (1)

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Here, $\mathrm{P(M)=m, P(P)=p, P(C)=c}$
The probability of passing in at least one subject $\mathrm{=1-P(\bar{M} \bar{P} \bar{C)}}$
$\mathrm{ \therefore \quad \frac{75}{100}=1-(1-m)(1-p)(1-c) }$
or $\mathrm{ \quad \frac{3}{4}=m+p+c-m p-p c-m c+m p c }$ ...................(1)
The probability of passing in at least two subjects
$\mathrm{ =P(M P C)+P(M P \bar{C})+P(M \bar{P} C)+P(\bar{M} P C) \\ }$
$\mathrm{ \therefore \frac{1}{2}=m p c+m p(1-c)+m(1-p) c+(1-m) p c \\ }$
$\mathrm{ \therefore 2 m p c=m p+m c+p c-\frac{1}{2} }$ ..................................................(2)

The probability of passing in exactly two subjects

$\mathrm{ =P(M P \bar{C})+P(M \bar{P} C)+P(\bar{M} P C) . \\ }$
$\mathrm{ \therefore \quad \frac{2}{5}=m p(1-c)+m(1-p) c+(1-m) p c=m p+m c+p c-3 m p c}$
(2) and (3) $\mathrm{ \Rightarrow 2 m p c+\frac{1}{2}=\frac{2}{5}+3 m p c \quad or \quad m p c=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}}$

$\mathrm{\therefore \quad (2) \Rightarrow \frac{1}{5}=m p+m c+p c-\frac{1}{2} \quad}$ or $\mathrm{\quad m p+m c+p c=\frac{1}{5}+\frac{1}{2}=\frac{7}{10}.}$
$\mathrm{\therefore (1) \Rightarrow \frac{3}{4}=m+p+c-\frac{7}{10}+\frac{1}{10}$

$\mathrm{ \therefore \quad m+p+c=\frac{27}{20}}$ .

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