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A box contains R red balls, B blue balls, and no other balls. One ball is removed and set aside, and then a second ball is removed. On each draw, each ball in the box is equally likely to be removed. The probability that both of these balls are red is 2/7 . The probability that exactly one of these balls is red is 1/2 . Determine the pair (R, B).

 Nov 9, 2019
 #1
avatar+2850 
+1
 
 

I tried to solve it, but it was too hard. This was my attempt, can someone tell me what I did wrong?

 

The probability that one red ball is selected:

\(\frac{r}{r+b}\).

With \(r\) being the number of red balls over \((r+b)\) , the total number of balls.

 

The probability that one blue ball is selected is similar:

\(\frac{b}{r+b}\).

 

 

Ok, so the problem states: "The probability that both of these balls are red is 2/7"

Lets make an equation for this:

 

\(\frac{r}{r+b}*\frac{r}{r+b}=\frac{2}{7}\)

 

The problem also states: "The probability that exactly one of these balls is red is 1/2"

Lets make an equation for this:

 

\(\frac{r}{r+b}*\frac{b}{r+b}=\frac{1}{2}\)

 

 

 

So now we have a system of equations:

\(\frac{r}{r+b}*\frac{r}{r+b}=\frac{2}{7}\)

\(\frac{r}{r+b}*​​\frac{b}{r+b}=\frac{1}{2}\) 

 

We solve:(Simplify fraction multiplications)

\(\frac{r^2}{(r+b)^2}=\frac{2}{7}\)

\(\frac{rb}{(r+b)^2}=\frac{1}{2}\)

 

Now cross multiply:

\(7r^2=2(r+b)^2\)

\(2rb=(r+b)^2\)

 

Simplify:

\(\frac{7r^2}{2}=(r+b)^2\)

\(2rb=(r+b)^2\)

 

After this I got stuck, I determined the ratio between r and b, which is \(7r=4b\).

 

HelP!

 
 Nov 9, 2019
 #2
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+1

Well, If you have R red balls out of a total (R + B), then when you draw the first one(assuming it is R), then don't you have left (R - 1) / (R + B - 1) for the 2nd draw?. Use "concrete numbers" to illustrate the point: Suppose you have 10 Red ball and 15 Blue balls. The first draw will be: 10 / [10+15]=10/25. The 2nd draw will have: 10 - 1/25 -1 =9/24. Then the probability of 2 Red ball will be: 10/25 x 9/24.

 Nov 9, 2019
 #3
avatar+2850 
+1

oohhhhh, ok

 

guest's way of determining probability + my strategy of solving using equations = your answer

CalculatorUser  Nov 9, 2019
 #4
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+1

CU: Try these numbers:

 

a=4,    b=3>>>>>>>    4/7 * 3/6 =12/42 = 2/7
a=12,  b= 10>>>>>>   12/22 * 11/21 =132 / 462 = 2/7

 Nov 9, 2019

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