1. **State the problem:** We are given two independent events A and B with $P(A) = 3P(B)$ and $P(A \cup B) = 0.68$. We need to find $P(B)$. 2. **Recall the formula for the union of two events:** $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ 3. **Use independence:** Since A and B are independent, $$P(A \cap B) = P(A) \times P(B)$$ 4. **Substitute $P(A) = 3P(B)$ into the union formula:** $$0.68 = 3P(B) + P(B) - 3P(B) \times P(B)$$ 5. **Simplify the equation:** $$0.68 = 4P(B) - 3P(B)^2$$ 6. **Rewrite as a quadratic equation:** $$3P(B)^2 - 4P(B) + 0.68 = 0$$ 7. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, $$P(B) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-4$, $c=0.68$. 8. **Calculate the discriminant:** $$\Delta = (-4)^2 - 4 \times 3 \times 0.68 = 16 - 8.16 = 7.84$$ 9. **Calculate the roots:** $$P(B) = \frac{4 \pm \sqrt{7.84}}{6} = \frac{4 \pm 2.8}{6}$$ 10. **Evaluate both solutions:** - $$P(B) = \frac{4 + 2.8}{6} = \frac{6.8}{6} = 1.1333...$$ (not possible since probability cannot exceed 1) - $$P(B) = \frac{4 - 2.8}{6} = \frac{1.2}{6} = 0.2$$ 11. **Final answer:** $$\boxed{P(B) = 0.2}$$