Question

The probabilities that a student passes in mathematics, physics and chemistry are $m,p$ and $c$ respectively. Of these subjects, a 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 subjects. Which of the following relations are true?

• A. $p+m+c=\frac{19}{20}$
• B. $p+m+c=\frac{27}{20}$
• C. $pmc=\frac{1}{10}$
• D. $pmc=\frac{1}{4}$

Answer 1

Answer: (B), (C)

The correct options are
B $p+m+c=\frac{27}{20}$
C $pmc=\frac{1}{10}$

Referring to the illustration:

Let s, t, u, v, w, x, y and z are the probabilities of the respective regions.

Thus, we have:

$s+t+u+v+w+x+y=0.75$ ...(i)

$v+w+x+y=0.5$ ...(ii)

$v+w+x=0.4$ ...(iii)

Thus, subtracting (ii) and (iii): $y=pmc=0.1=\frac{1}{10}$

Here, y refers to the intersection of all the three and hence equals $p\ast c\ast m$ since they are independent events.

Again, subtracting (i) and (ii): $s+t+u=0.25$

We need to find $m+c+p$:

$m+c+p=s+t+u+2(v+w+x)+3y=0.25+2\ast 0.4+3\times 0.1=1.35=\frac{27}{20}$
Hence, (b), (c) are correct.