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. Which of the following relations are true?

• A. $p+m+c=19/20$
• B. $p+m+c=27/20$
• C. $pmc=1/10$
• D. $pmc=1/4$

Answer 1

Answer: (B), (C)

$pmc=1/10$
$p+m+c=27/20$

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):

$y=p\times m\times c=0.1$

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

Again, subtracting (i) and (ii):

$s+t+u=0.25$

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

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