Alright, let's break down the given problem step-by-step to find the required probabilities.
### 1. Describing the Contents of Each Urn
We have:
- Silver Urn: 8 black balls and 9 brown balls.
- Golden Urn: 4 red balls, 3 yellow balls, and 3 blue balls.
### 2. Calculating the Total Number of Balls in Each Urn
First, let's calculate the total number of balls in each urn:
- Silver Urn: [tex]\(8 + 9 = 17\)[/tex] balls.
- Golden Urn: [tex]\(4 + 3 + 3 = 10\)[/tex] balls.
### 3. Finding Individual Probabilities
#### Probability of Picking Each Type of Ball from the Silver Urn
- Probability of picking a black ball from the silver urn:
[tex]\[ \text{P(Black)} = \frac{8}{17} \][/tex]
- Probability of picking a brown ball from the silver urn:
[tex]\[ \text{P(Brown)} = \frac{9}{17} \][/tex]
#### Probability of Picking Each Type of Ball from the Golden Urn
- Probability of picking a red ball from the golden urn:
[tex]\[ \text{P(Red)} = \frac{4}{10} = 0.4 \][/tex]
- Probability of picking a yellow ball from the golden urn:
[tex]\[ \text{P(Yellow)} = \frac{3}{10} = 0.3 \][/tex]
### 4. Calculating Combined Probabilities
#### Probability of Picking a Black Ball from the Silver Urn and a Red Ball from the Golden Urn
[tex]\[ \text{P(Black and Red)} = \left(\frac{8}{17}\right) \times 0.4 \][/tex]
#### Probability of Picking a Brown Ball from the Silver Urn and a Yellow Ball from the Golden Urn
[tex]\[ \text{P(Brown and Yellow)} = \left(\frac{9}{17}\right) \times 0.3 \][/tex]
### 5. Adding the Combined Probabilities
To answer the question, we need the total probability of either event occurring:
[tex]\[ \text{Total Probability} = \text{P(Black and Red)} + \text{P(Brown and Yellow)} \][/tex]
### 6. Results
- The probability that we get a black ball from the silver urn and a red ball from the golden urn is:
[tex]\[ \boxed{0.188} \][/tex]
- The probability that we get a brown ball from the silver urn and a yellow ball from the golden urn is:
[tex]\[ \boxed{0.159} \][/tex]
- The total probability that either of these events occurs is:
[tex]\[ \boxed{0.347} \][/tex]
Thus, the final answer to the question about the probability of either getting a black ball from the silver urn and a red ball from the golden urn, or a brown ball from the silver urn and a yellow ball from the golden urn, is [tex]\(0.347\)[/tex] rounded to three decimal places.
