GATE CSE 1995 | Question: 2.14

Question

A bag contains $10$ white balls and $15$ black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is:

• A. $\frac{2}{3}$

• B. $\frac{4}{5}$

• C. $\frac{1}{2}$

• D. $\frac{1}{3}$

Comments

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By Three Method : https://gateoverflow.in/2626/gate-cse-1995-question-2-14?show=537763#a537763

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people will not realize why the multiplication happened, because they will not use conditional probability,

here's why the multiplication is used:

We are asked what is the probability of white and black ball drawn in succession,

let me take first white is drawn and then black the same can be understood for black then white.

now P(white intersection black) is asked but why intersection? because it's said a white and a black ball is drawn "and", means if both events are necessary to happen then they must happen together they need to intersect together.

 

and now,

P(white intersection black) = say I know the probability of ball being white that is P(W) then I need to know probability of ball being black given a white ball is drawn that is P(B|W)

so P(W intersection Black) = P(W) * P(B|W)

and we have P(W) and we have P(B|W) we can calculate this now.

Answer 1

Answer is C

• Probability of first ball white and second one black $=\left(\dfrac{10}{25}\right)\times \left(\dfrac{15}{24}\right)$
• Probability of first ball black and second one white$=\left(\dfrac{15}{25}\right)\times \left(\dfrac{10}{24}\right)$
• Probability $=$ sum of above two probabilities $=\dfrac{1}{2}.$

Comments

(10C1 * 15C1)/(25C2)=0.5(hypergeometric distribution)

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👌 👍

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All the below solution though somehow lead to right answer but the appraoch is wrong this is correct approach because its said 2 succesive draws so firstly we can either pick one black ball or red ball and then in next draw number of balls will decrease so next picking the below denominator will become 24 all the solutions below solve the question " what is the probablitly of picking a black and white ball from total balls" though leading to coprrect answer the intution and approach is this one best answer

Answer 2

SInce 2 balls are drawn, sample space S = 2^2 = {BB,WW,BW,WB}

Probability that we get one of balck and one of white = (BW,WB)/S = 2/4 = 1/2

Answer : (c)

Comments

So according to you, if there are 25 white balls and 15 black ,then also ans will be 1/2 ?

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yes this is not the right way to do here

Answer 3

Probability that both balls are black=  (15C2)/(25C2) =7/20

Probability that both balls are white=  (10C2)/(25C2) =6/40

Probability that (both balls are black) or (both balls are white)= 7/20 + 6/40 =20/40=1/2

The probability that one of them is black and the other is white is: 1- 1/2 = 1/2

Answer 4

Probability = (10C1 * 15C1)/25C2

Answer 5

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Answer 6

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