An urn contains 8 green and 6 pink balls. Six balls are randomly drawn from the urm in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 6 balls drawn from the um are green? Round your answer to three decimal places.

To solve the problem, let's break it down step by step.

### Step 1: Understand the Scenario
- There are a total of \(8\) green balls and \(6\) pink balls in the urn.
- The total number of balls in the urn is \(8 + 6 = 14\).
- The drawing is done with replacement, meaning after each draw, the ball is put back into the urn.

### Step 2: Probability of Drawing a Green Ball
The probability of drawing a green ball on any single draw is the ratio of green balls to the total number of balls:

\[
P(\text{Green}) = \frac{8}{14} = \frac{4}{7}
\]

### Step 3: Probability of Drawing 6 Green Balls in Succession
Since the balls are drawn with replacement, each draw is independent of the previous ones. Therefore, the probability of drawing 6 green balls in succession is the product of the individual probabilities:

\[
P(\text{6 Green Balls}) = \left(\frac{4}{7}\right)^6
\]

### Step 4: Calculate the Probability
Now, we calculate the above expression:

\[
P(\text{6 Green Balls}) = \left(\frac{4}{7}\right)^6 = \frac{4096}{117649} \approx 0.0348
\]

### Final Answer
The probability that all 6 balls drawn are green is approximately \(0.035\) when rounded to three decimal places.

Would you like further details or have any questions?

Here are 8 related questions you might find interesting:
1. What is the probability of drawing exactly 3 green balls and 3 pink balls?
2. How does the probability change if the balls are drawn without replacement?
3. What is the probability that none of the drawn balls are green?
4. How many different sequences of balls could result in exactly 4 green balls being drawn?
5. What if the urn contained 10 green and 10 pink balls? How would the probability change?
6. What is the expected number of green balls in 6 draws?
7. How does the probability distribution look like for different numbers of green balls drawn?
8. What is the variance in the number of green balls drawn in 6 attempts?

**Tip:** When dealing with probabilities, especially with replacement scenarios, remember that each draw is independent of the others, which greatly simplifies the calculations.

Calculate the probability of drawing all 6 green balls from an urn containing 8 green and 6 pink balls with replacement. Understand the concept of independent events and use the probability formula for successive draws. Round your answer to three decimal places.

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