1. **State the problem:** We have a bag with red and blue counters in the ratio 4 : 10. 2. **What is asked:** Find the probability that a randomly chosen counter is blue. 3. **Recall the formula for probability:** $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$ 4. **Identify favorable outcomes:** The favorable outcomes are the blue counters, which are 10 parts. 5. **Calculate total parts:** Total counters = red + blue = 4 + 10 = 14 parts. 6. **Calculate probability:** $$\text{Probability(blue)} = \frac{10}{14}$$ 7. **Simplify the fraction:** Both numerator and denominator can be divided by 2: $$\frac{10 \div 2}{14 \div 2} = \frac{5}{7}$$ 8. **Final answer:** The probability that a randomly chosen counter is blue is $$\boxed{\frac{5}{7}}$$.