The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has a 75% chance of passing in at least one, a 50% chance of passing in at least two and a 40% chance of passing in exactly two. Which of the following relations are true ?

# The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has a 75% chance of passing in at least one, a 50% chance of passing in at least two and a 40% chance of passing in exactly two. Which of the following relations are true ?

1. A

$\mathrm{p}+\mathrm{m}+\mathrm{c}=\frac{19}{20}$

2. B

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

3. C

$\mathrm{pmc}=\frac{1}{10}$

4. D

$\mathrm{pmc}=\frac{1}{4}$

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### Solution:

$\begin{array}{l}\mathrm{P}\left(\mathrm{m}\cup \mathrm{p}\cup \mathrm{c}\right)=\frac{75}{100}\\ ⇒\mathrm{m}+\mathrm{p}+\mathrm{c}-\sum \mathrm{mp}+\mathrm{mpc}=\frac{3}{4}\to \left(1\right)\end{array}$
p(passing at least 2 subjects)
$=\sum \mathrm{mp}-2\mathrm{mpc}=\frac{50}{100}=\frac{1}{2}\to \left(2\right)$
p(passing exactly 2 subjects)
$=\sum \mathrm{mp}-3\mathrm{mpc}=\frac{40}{100}=\frac{2}{5}\to \left(3\right)$
(2) and (3) gives $\mathrm{mpc}=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}$
From (2)$\sum \mathrm{mp}=\frac{1}{2}+2\mathrm{mpc}=\frac{1}{2}+\frac{1}{5}=\frac{7}{10}$
From$\sum \mathrm{m}=\frac{3}{4}+\sum \mathrm{mp}-\mathrm{mpc}$
$=\frac{3}{4}+\frac{7}{10}-\frac{1}{10}=\frac{15+14-2}{20}=\frac{27}{20}$

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