MHB What is the number of blue balls in the second urn?

AI Thread Summary
The problem involves two urns: the first contains 4 red and 6 blue balls, while the second has 16 red balls and an unknown number of blue balls. The probability that both balls drawn from each urn are of the same color is given as 0.44. The correct approach to solve this involves calculating the probabilities for both colors and setting up the equation: (4/10)(16/(16+x)) + (6/10)(x/(16+x)) = 0.44. Solving this equation reveals that the number of blue balls in the second urn is 4.
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Question:
An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is .44. Calculate the number of blue balls in the second urn.

My attempt
I don't know the best notation to use for this situation. I am going to try [math]P(R_1)[/math] as the probability of drawing a red ball from the first urn. So we have [math]P(R_1)=.4 \text{ and } P(B_1)=.6[/math]. To express the probability of both balls being a single color it seems there are two cases to consider which we should add: [math]P(R_1 \cap R_2)+P(B_1 \cap B_2)[/math]. Am I correct in thinking that for mutually exclusive events that's the same as [math]P(R_1 \cdot R_2)+P(B_1 \cdot B_2)[/math]?

I know the basic ways to manipulate these using DeMorgan's Laws but I'm missing the first step or have set up the problem entirely incorrectly. I have the solution key but I don't want the full solution yet.
 
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Houdini said:
Question:
An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is .44. Calculate the number of blue balls in the second urn.
I would solve the following:
\frac{4}{10}\frac{16}{16+x}+\frac{6}{10}\frac{x}{16+x}=\frac{44}{100}.
 
Makes perfect sense. I've been trying to use all of these set rules that I missed it. So x=4 and plugging that in confirms.
 
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