Question

The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. On 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.

• A.
• B.
• C.
• D.

Answer 1

Answer: (B), (C)

The correct options are
B
C

Let M, P and C be the events of passing in mathematics, physics and chemistry respectively.
$P(M\cup P\cup C)=\frac{75}{100}=\frac{3}{4}$
$P(M\cap P)+P(P\cap C)+P(M\cap C)-2P(M\cap P\cap C)=\frac{50}{100}=\frac{1}{2}$
$P(M\cap P)+P(P\cap C)+P(M\cap C)-3P(M\cap P\cap C)=\frac{40}{100}=\frac{2}{5}$
$\therefore m(1-p)(1-c)+p(1-m)(1-c)+c(1-m)(1-p)+mp(1-c)+mc(1-p)+pc(1-m)+mpc=\frac{3}{4}$
$\Rightarrow m+p+c-mc-mp-pc+mpc=\frac{3}{4}.....\left(i\right)$
Similarly, $mp(1–c)+pc(1–m)+mc(1–p)+mpc=\frac{1}{2}$ $\Rightarrow mp+pc+mc–2mpc=\frac{1}{2}$ .....(ii)
$mp(1–c)+pc(1–m)+mc(1–p)=\frac{2}{5}$
$\Rightarrow mp+pc+mc–3mpc=\frac{2}{5}$ ......(iii)
From (ii) to (iii), $mpc=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}$
From (i) and (ii), $m+p+c–mpc=\frac{3}{4}+\frac{1}{2}$ $\therefore m+p+c=\frac{3}{4}+\frac{1}{2}+\frac{1}{10}=\frac{15+10+2}{20}=\frac{27}{20}$