TIFR CSE 2012 | Part A | Question: 20

Question

There are $1000$ balls in a bag, of which $900$ are black and $100$ are white. I randomly draw $100$ balls from the bag. What is the probability that the $101$st ball will be black?

• A. $9/10$
• B. More than $9/10$ but less than $1$.
• C. Less than $9/10$ but more than $0$.
• D. $0$
• E. $1$

Comments

How ratio will remain same throughout the event? Since we are picking the balls the number of balls left in bag is changing, so the ratio of black: white ball will change everytime we pick a ball..

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Rather than picking up all the 100 balls at once, if we pick 100 balls one by one(one ball at a time), then the probability will be less than 9/10, right? Because 9/10 is based on initial 900 black and 100 red balls. But while picking one by one, the decrease in number of black balls has been more than the decrease in number of red balls. So, by time the probability 9/10 will decrease.

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Someone might get confused that why answer is not in range let me tell you with respect to “Theorem of Total Probability” (just a fancy word for something very intuitive)

P(101th ball is black) = P[101th ball is black/(You draw 100 white balls and 0 black balls previously)] + P[101th ball is black/(You draw 99 white balls and 1 black balls previously)] + ………..

Now this is the reason that answer wont be in a range, also instead of doing so rigorous math its better to use expectation which is used in the below answer

Hope this helps

Answer 1

Among 1000 balls 900 black balls can be choosen in 9/10 ways

so, Ans will be A

Comments

yes, answer is 9/10. And it is the same for even 800th draw. Question is just for checking the correct concept.

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ok, I have edited it

yes , there will be some more case if I check individually, corrected now :)

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@srestha ma'am,  you have mentioned that you have considered the worst case scenario  of getting first 100 ball is white, but if we take this then we get probability of black ball in next draw will be 1(bz we only left with black ball)'

please guide this