Question

The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has 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. Following relations are drawn in m, p, c

$I.p+m+c=\frac{27}{20}II.p+m+c=\frac{13}{20}III.\left(p\right)\times \left(m\right)\times \left(c\right)=\frac{1}{10}$

• A. Only relation I is true.
• B. Only relation III is true.
• C. Relations II and III are true.
• D. Relations I and III are true.

Answer 1

The correct option is D Relations I and III are true.



$\text{Given},\alpha +\beta +\gamma +d+e+f+g=\frac{75}{100}$ ...(1)

& $d+e+f+g=\frac{50}{100}$ ...(2)

& $d+e+f=\frac{40}{100}$ ...(3)

Subtracting (2) & (3) $g=\frac{10}{100}$ = 10%
& subtracting (1) & (2),

$\alpha +\beta +\gamma =\frac{25}{100}$

So M × P × Ch = 10%

Now, M + P + Ch

$\Rightarrow (\alpha +d+g+f)+(\beta +d+e+g)+(\gamma +e+f+g)$
$=(\alpha +\beta +\gamma )+2(d+e+f)+3g$

$=\frac{25}{100}+2\left(\frac{40}{100}\right)+3\left(\frac{10}{100}\right)$

$=\frac{135}{100}=\frac{27}{20}$

Method - II
% of students passing in atleast one subject = 75%

$\text{i.e.}p+m+c-\{p\cap m+m\cap c+c\cap p\}+p\cap m\cap c=0.75$ ...(1)

% of students passing in at least two subjects = 50%

$\{p\cap m+m\cap c+c\cap p\}-2\{p\cap m\cap c\}=0.5$ ...(2)

% of students passing in exactly two subjects = 40%

$\{p\cap c+m\cap c+c\cap p\}-3\{p\cap m\cap c\}=0.4$ ...(3)

By (2) and (3)

$p\cap m\cap c=0.1\Rightarrow p\times m\times c=\frac{1}{10}$

Adding (1) and (2)

$p+m+c-(p\cap m\cap c)=1.25$

$p+m+c=1.25+0.1=1.35=\frac{135}{100}=\frac{27}{20}$

Hence option (d) is correct