An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

First, we find the probability both are green, then the probability both are blue, and add the two probabilities. The sum should be equal to $0.58$.
The probability both are green is $\frac{4}{10}\cdot\frac{16}{16+N}$, and the probability both are blue is $\frac{6}{10}\cdot\frac{N}{16+N}$, so\[\frac{4}{10}\cdot\frac{16}{16+N}+\frac{6}{10}\cdot\frac{N}{16+N}=\frac{29}{50}\]Solving this equation,\[20\left(\frac{16}{16+N}\right)+30\left(\frac{N}{16+N}\right)=29\]Multiplying both sides by $16+N$, we get\[20\cdot 16 + 30\cdot N = 29(16+n)\Rightarrow 320+30N=464+29N \Rightarrow N = \boxed{144}\]
144
Level 5
Counting & Probability
Problem: An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the value of N, we first need to determine the total number of balls in each urn. The first urn has 4 green and 6 blue balls, totaling 10 balls. The second urn has 16 green balls and N blue balls, totaling 16 + N balls.

Next, we calculate the probability of drawing two balls of the same color. This can happen in two scenarios: both balls are green or both balls are blue.

1. **Probability of both balls being green**:
   The probability of drawing a green ball from the first urn is 4/10. The probability of drawing a green ball from the second urn is 16/(16 + N). Therefore, the combined probability for both being green is:
   (4/10) * (16/(16 + N)) = (4 * 16) / (10 * (16 + N)) = 64 / (10 * (16 + N)).

2. **Probability of both balls being blue**:
   The probability of drawing a blue ball from the first urn is 6/10. The probability of drawing a blue ball from the second urn is N/(16 + N). Thus, the combined probability for both being blue is:
   (6/10) * (N/(16 + N)) = (6 * N) / (10 * (16 + N)).

Now, we add these two probabilities to find the total probability of both balls being the same color:
Total probability = (64 / (10 * (16 + N))) + (6N / (10 * (16 + N))) = (64 + 6N) / (10 * (16 + N)).

We know from the problem statement that this total probability equals 0.58:
(64 + 6N) / (10 * (16 + N)) = 0.58.

To eliminate the fraction, we cross-multiply:
64 + 6N = 0.58 * 10 * (16 + N).

Calculating the right side:
0.58 * 10 = 5.8,
thus,
64 + 6N = 5.8 * (16 + N).

Expanding the right side:
64 + 6N = 92.8 + 5.8N.

Now, we isolate N:
64 + 6N - 5.8N = 92.8,
0.2N = 92.8 - 64,
0.2N = 28.8.

Dividing both sides by 0.2 gives:
N = 28.8 / 0.2 = 144.

Thus, the value of N is 144.<|END_OF_LATENT|> The final answer is $144$.