Choosing Monochromatic Marbles

Probability Level 3

There are two boxes which contain both black and white marbles. The total number of marbles in the two boxes is 25 . When one marble is taken out of each box randomly, the probability that both marbles are black is $\frac{27}{50}$ ,

When one marble is taken out of each box randomly, the probability that both marbles are white is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

Image credit: Flickr Ksionic

The answer is 26.

5 solutions

Josh Speckman
Apr 19, 2014

First, we can find how many marbles are in each box. If there are $x$ marbles in box $1$ and $y$ marbles in box $2$ , we must have $x+y=25$ and $xy$ is a multiple of $50$ . Using these guidelines, we see that $x=5, y=20$ is a solution. If there are $a$ black marbles in box $1$ and $b$ marbles in box $2$ , we have $ab=54$ . Since there cannot be more than $5$ marbles in box $1$ or $20$ marbles in box $2$ , we find there are $3$ black marbles in box $1$ and $18$ in box $2$ . Indeed, $\dfrac{3}{5} \cdot \dfrac{18}{20} = \dfrac{27}{50}$ , so this is the correct pairing. The probability of drawing $2$ white marbles is $\dfrac{2}{5} \cdot \dfrac{2}{20} = \dfrac{2}{5} \cdot \dfrac{1}{10} = \dfrac{1}{25}$ , and $1 + 25 = \fbox{26}$

Can't we solve it like this??.......If probability of black is 27/50 , then probability of not black(means white) will be 1-(27/50), hence m+n will be 73!

Ashu Jain - 7 years, 1 month ago

23/50 is for the two different cases ( black , white ) and ( white , white )

Tan Li Xuan - 7 years, 1 month ago

in that case x=15 ,y=10 is also a solution,p(both black)=9 9/150=27/50.therefore p(both white)=(15-9)/15 (10-9)/10=1/25. therefore m+n=26 leads to same result.

MOHD FARAZ - 7 years, 1 month ago

Jon Haussmann
Apr 19, 2014

This is from the 2000 AIME I .

Vipul Soni
Apr 25, 2014

2 Solutions are possible

both boxes contain 9 black marbles , with the total being 10 &15, and

3 & 18 black marbles with total of 5 & 20 marbles respectively in each box.

both give answer m=1, n=25.

Hence, m+n=26

Hadia Qadir
Jul 31, 2015

The probability of drawing two black marbles is 0.54. In order to achieve this probability, there are two commutative options: 1) P(Box 1 = Black) = 1.0 and P(Box 2 = Black) = 0.54 2) P(Box 1 = Black) = 0.9 and P(Box 2 = Black) = 0.6 Since we are working in the discrete case of marbles, option #1 is unobtainable because we cannot achieve a discrete number of less than 25 marbles with a probability of 0.54. Thus option #2 is the only solution. To find the probability of selecting a white marble from each box, simply take (1 - 0.9) x (1 - 0.6) = 0.04 = 1/25. So 1 + 25 = 26

Christopher Ho
Apr 25, 2014

The probability of drawing two black marbles is 0.54. In order to achieve this probability, there are two commutative options:

1) P(Box 1 = Black) = 1.0 and P(Box 2 = Black) = 0.54

2) P(Box 1 = Black) = 0.9 and P(Box 2 = Black) = 0.6

Since we are working in the discrete case of marbles, option #1 is unobtainable because we cannot achieve a discrete number of less than 25 marbles with a probability of 0.54.

Thus option #2 is the only solution. To find the probability of selecting a white marble from each box, simply take (1 - 0.9) x (1 - 0.6) = 0.04 = 1/25. So 1 + 25 = 26

Choosing Monochromatic Marbles

Image credit: Flickr Ksionic

The answer is 26.

5 solutions

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