GATE IT 2005 | Question: 1

Question

A bag contains $10$ blue marbles, $20$ green marbles and $30$ red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag. This process is repeated $3$ times. The probability that no two of the marbles drawn have the same colour is

• A. $\left(\dfrac{1}{36}\right)$
• B. $\left(\dfrac{1}{6}\right)$
• C. $\left(\dfrac{1}{4}\right)$
• D. $\left(\dfrac{1}{3}\right)$

Comments

Notice ->

Direct Hypergeometric distribution formula can not be applied because we are placing marbel back in the back.

Answer 1

No two marbles have the same color means, the final outcome of the three draws must be a permutation of

Blue, Green, Red

There are $3! = 6$ such permutations possible.

Now, probability of getting a Blue first, Green second and Red third $=\dfrac{10}{60}\times \dfrac{20}{60}\times \dfrac{30}{60}$

Required probability $=6\times \dfrac{10}{60}\times \dfrac{20}{60}\times \dfrac{30}{60} =\dfrac{1}{6}.$

Correct Answer: $B$

Comments

if we use (10C1 +20C2 + 30C2 ) / 60C2 is it ok ? in this one we have to mutilplay 3! or this formula give direct  right answer ?

formula is nCr

please guide me is it right or not ?

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here in the question order of drawing a marble is not mentioned so we need to take the posibilities of orders so thats why we are multiplying with 3!

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completely wrong , answer is 2/9

total combinations with three colors b,g,r is 27

favorable are 6 bgr,brg,rgb,rbg,grb,gbr

so 6/27=2/9

Answer 2

P (no two marble have the same colour) = 1 - P(Two or more marble have the same colour)

Case I --> P( Two marble same colour) = 3C2*P(both Red ,blue or Green) = 3*{(1/2*1/2*1/2)+(1/6*1/6*5/6)+(1/3*1/3*2/3)} ==> 24/36

Case II--> P(Three marble same colour) = 3C3*P(All Red, Blue or Green) = 1*{(1/2*1/2*1/2)+(1/3*1/3*1/3)+(1/6*1/6*1/6)} ==> 6/36

Therefore Probability of no two drawn marble have the same colour ==> { 1 - (30/36)} = 1/6 .

Comments

You have arrived at correct answer. But for going from Delhi to Calcutta, instead of taking domestic flight you have taken international flight, flying over Atlantic and Pacific.

Answer 3

10/20 * 20/60 * 30/60 =1/6

Comments

what's wrong in it????plz explain

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1/6 is correct. But calculation is wrong.