1. **Problem Statement:** We are given the function $$f(x) = 1 - \left(1 - \frac{x}{102}\right)^{\frac{1}{6}}$$ defined for $$0 \leq x \leq 120$$. We want to find the probability that a life aged 35 dies before age 55. 2. **Understanding the problem:** The function $$f(x)$$ represents a cumulative distribution function (CDF) for the age at death starting from age 0. Here, $$x$$ is the age. 3. **Adjusting for the given age:** Since the person is currently 35 years old, we want the probability that death occurs before age 55, which corresponds to the interval from 35 to 55. 4. **Formula for conditional probability:** The probability that a life aged 35 dies before age 55 is given by $$P(35 < X < 55 \mid X > 35) = \frac{F(55) - F(35)}{1 - F(35)}$$ where $$F(x) = f(x)$$ is the CDF. 5. **Calculate $$F(35)$$:** $$F(35) = 1 - \left(1 - \frac{35}{102}\right)^{\frac{1}{6}} = 1 - \left(1 - 0.343137\right)^{\frac{1}{6}} = 1 - (0.656863)^{\frac{1}{6}}$$ Calculate $$0.656863^{1/6}$$: $$0.656863^{1/6} = e^{\frac{1}{6} \ln(0.656863)} \approx e^{\frac{1}{6} \times (-0.420)} = e^{-0.07} \approx 0.9324$$ So, $$F(35) = 1 - 0.9324 = 0.0676$$ 6. **Calculate $$F(55)$$:** $$F(55) = 1 - \left(1 - \frac{55}{102}\right)^{\frac{1}{6}} = 1 - \left(1 - 0.539216\right)^{\frac{1}{6}} = 1 - (0.460784)^{\frac{1}{6}}$$ Calculate $$0.460784^{1/6}$$: $$0.460784^{1/6} = e^{\frac{1}{6} \ln(0.460784)} \approx e^{\frac{1}{6} \times (-0.776)} = e^{-0.1293} \approx 0.8787$$ So, $$F(55) = 1 - 0.8787 = 0.1213$$ 7. **Calculate the conditional probability:** $$P = \frac{F(55) - F(35)}{1 - F(35)} = \frac{0.1213 - 0.0676}{1 - 0.0676} = \frac{0.0537}{0.9324} \approx 0.0576$$ 8. **Interpretation:** The probability that a life aged 35 dies before age 55 is approximately $$0.0576$$ or 5.76%.