To solve this problem, we'll use probability concepts, particularly focusing on combinations and dependent events.

We are asked to find the probability that all four people selected are in their 50s. Here's the step-by-step process:

1. Total number of people: $10 \text{ (30-39)} + 20 \text{ (40-49)} + 30 \text{ (50-59)} + 15 \text{ (60-69)} = 75 \text{ people in total}.$

2. Number of people in their 50s: $30 \text{ people in the 50-59 age group}.$

3. Selecting 4 people out of 75: The total number of ways to select 4 people from the 75 is given by combinations: $\binom{75}{4} = \frac{75 \times 74 \times 73 \times 72}{4 \times 3 \times 2 \times 1} = 421,200.$

4. Selecting 4 people from the 50s group: The number of ways to select 4 people from the 30 people in their 50s is: $\binom{30}{4} = \frac{30 \times 29 \times 28 \times 27}{4 \times 3 \times 2 \times 1} = 27,405.$

5. Probability calculation: The probability that all four selected people are in their 50s is: $P = \frac{\binom{30}{4}}{\binom{75}{4}} = \frac{27,405}{421,200} \approx 0.065.$

Thus, the probability that all four people selected are in their 50s is approximately $\mathbf{0.065}$ to three decimal places.

Would you like more details or have any questions?

Here are some related questions:

1. What is the probability of selecting exactly 3 people in their 50s?
2. What is the probability of selecting no one in their 50s?
3. How would the probability change if the total group size increased to 100?
4. What is the probability of selecting one person from each age group?
5. How does the number of people in the 50s age group affect the probability?

Tip: In problems involving selection without replacement, combinations are used to calculate the total number of possible outcomes and favorable outcomes.