1. **State the problem:** Calculate the value of the expression $$\frac{1.96^2 \times 0.5 \times (1 - 0.5)}{0.05^2}$$. 2. **Write the formula and explain:** This is a fraction where the numerator is the product of $1.96^2$, $0.5$, and $(1 - 0.5)$, and the denominator is $0.05^2$. 3. **Calculate the numerator:** $$1.96^2 = 3.8416$$ $$1 - 0.5 = 0.5$$ So, $$3.8416 \times 0.5 \times 0.5 = 3.8416 \times 0.25 = 0.9604$$ 4. **Calculate the denominator:** $$0.05^2 = 0.0025$$ 5. **Form the fraction:** $$\frac{0.9604}{0.0025}$$ 6. **Divide numerator by denominator:** $$\frac{\cancel{0.9604}}{\cancel{0.0025}} = 384.16$$ 7. **Final answer:** $$384.16$$ This means the value of the given expression is 384.16.