To determine the smallest possible number of red counters in the bag, let’s analyze the given probabilities and apply the concepts accordingly.

1. Set up the information from the probabilities:

   • The probability of choosing a black counter is $0.7$.
   • The probability of choosing a white counter is $0.15$.
   • The probability of choosing a red counter is $0.15$.

2. Let $N$ represent the total number of counters in the bag. The probabilities are fractions of the total number, meaning:

   • The number of black counters would be $0.7 \times N$.
   • The number of white counters would be $0.15 \times N$.
   • The number of red counters would also be $0.15 \times N$.

3. Find the smallest $N$ such that each probability multiplied by $N$ results in a whole number:

   • Since $0.7$, $0.15$, and $0.15$ are fractions with denominators that suggest multiples of 10, $N$ must be a multiple of 10 for each calculation to yield whole numbers.

4. Calculate the counts for $N = 10$ (smallest possible multiple of 10):

   • Number of black counters: $0.7 \times 10 = 7$
   • Number of white counters: $0.15 \times 10 = 1.5$ (not a whole number)

   Since $N = 10$ does not yield whole numbers for all colors, try the next smallest multiple of 10, which is $N = 20$.

5. Calculate the counts for $N = 20$:

   • Number of black counters: $0.7 \times 20 = 14$
   • Number of white counters: $0.15 \times 20 = 3$
   • Number of red counters: $0.15 \times 20 = 3$

   For $N = 20$, all counts are whole numbers: 14 black, 3 white, and 3 red counters.

6. Conclusion: The smallest possible number of red counters in the bag is 3.

Would you like more details on any part of this solution?

Here are five questions for further exploration:

1. What if the probability of black counters was slightly different, say 0.6? How would that change $N$?
2. How can we generalize this solution for any probability distribution?
3. What if we wanted to maximize the number of red counters instead of minimizing it?
4. How does this concept relate to the probability of other outcomes in more complex systems?
5. What are some other real-world examples where we might use similar probability calculations?

Tip: When dealing with probabilities in practical problems, ensuring that all quantities align as whole numbers often involves finding a least common multiple.