GATE2012 AR: GA-9

Question

A smuggler has $10$ capsules in which five are filled with narcotic drugs and the rest contain the original medicine. All the $10$ capsules are mixed in a single box, from which the customs officials picked two capsules at random and tested for the presence of narcotic drugs. The probability that the smuggler will be caught is

• A. $0.50$
• B. $0.67$
• C. $0.78$
• D. $0.82$

Comments

What a question !

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C

Answer 1

The smuggler would be caught if any one or both the randomly picked capsule contains drugs.
$M$ represents Medicine, $D$ represents Drugs

• $A -$ The smuggler will be caught
• $B -$ The randomly picked capsules contains $M, D$
• $C -$ The randomly picked capsules contains $D, M$
• $D -$ The randomly picked capsules contains $D, D$

$P(A) = P(B) + P(C) + P(D)$

$P(A) = (5/10)\times (5/9) + (5/10)\times (5/9) + (5/10)\times(4/9)$
$\quad \quad = 0.278 + 0.278 + 0.222$
$\quad \quad =0.778$

Hence, Answer is Option (C) 0.78

Comments

Here you are using the fact that picking capsule one after the other.

But in question it was said directly to pick two capsules(not one after the other). So there shouldn't be three cases .

Please suggest if u am wrong.

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SANDEEP1729

yes the capsules are picked randomly and not one after the other.

Answer 2

There are 2 ways by which smuggler would be caught. 

1. Both capsules contain narcotic drugs
2. One of them contain narcotic drugs

Total combinations where both capsules contain narcotic drugs = $\large \binom{5}{2} = 10$

Total combinations where one of the two capsules contain narcotic drugs = $\large \binom{5}{1} \binom{5}{1} = 25$

Total number of combinations = $\large \binom{10}{2} = 45$

Probability = $\large \frac{10+25}{45} = \frac{7}{9} = 0.777777778$

Answer 3

$1 - \frac{\binom{5}{2}}{\binom{10}{2}} = 1- \frac{2}{9} = 0.777777 \simeq 0.78$

So answer is part (C).

Comments

Sir why subtract from 1 not getting

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He is doing reverse thing.i.e.

1 - P(picking capsule that do not contain drugs) = P(picking capsule containing drugs.)