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# help

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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
+2551
+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
+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
+2551
+1

oohhhhh, ok

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

CalculatorUser  Nov 9, 2019
#4
+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