Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A bag contains colored balls of which at least $90\%$ are red. Balls are drawn from the bag one by one and their color noted. It is found that $49$ of the first $50$ balls drawn are red. Thereafter $7$ out of every $8$ balls drawn are red. The number of balls in the bag CAN NOT BE

  1. $170$
  2. $210$
  3. $250$
  4. $194$

and the answer turns out to be $250$.

MY TRY: $\dfrac{\binom{0.9n}{1}\binom{0.9n-1}{1}\cdots\binom{0.9n-48}{1}\times\binom{0.1n}{1}}{\binom{n}{1}\binom{n-1}{1}\cdots\binom{n-48}{1}\times\binom{n-49}{1}}=\dfrac{49}{50}$ and similarly for 7 & 8.

How can we get the answer?

share|improve this question

2 Answers 2

up vote 6 down vote accepted

The total number of red balls drawn after $k$ subsequent draws is $49 + 7k$. The total number of balls drawn is $50 + 8k$.

The ratio of the is equal to the proportion of the red balls in the bag, which is at least $0.9$. In other words,

$$\frac{49 + 7k}{50 + 8k} \ge 0.9$$ $$49 + 7k \ge 45 + 7.2k$$ $$0.2k \le 4$$ $$k \le 20$$

Hence, the total number of balls drawn,

$$50 + 8k \le 50 + 20\cdot8 = 210$$

This says that at most $210$ balls can be drawn. So there must be at most $210$ balls in the bag, and so there cannot be $250$ balls in the bag (because $250 > 210$).

share|improve this answer
    
nice answer@yiyuan –  Semsem Apr 5 at 7:58
    
Thank you so much, Sir. So good answer! –  Sush Apr 5 at 7:59

Set up the ratio: $\frac{49 + 7x}{50 + 8x} = \frac{9}{10}$. Solving it explicitly gives you $x = 20$, so (2) is a valid answer.

If $x = 15$, we get $170$ as the denominator, and $90.6\%$.

When $x = 18$, we get $194$ as the denominator and $90.2\%$

When $x = 25$, we get 250 in the denominator and $89.6\%$.

share|improve this answer
    
Thank you so much :) –  Sush Apr 5 at 7:58

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.