1. **State the problem:** We are given conditional probability $P(B|A') = \frac{7}{10}$, $P(A') = \frac{3}{8}$, and $P(B) = \frac{2}{5}$. We need to find $P(A \cap B)$. 2. **Recall the formula for total probability:** $$P(B) = P(B|A)P(A) + P(B|A')P(A')$$ where $P(A) = 1 - P(A')$. 3. **Calculate $P(A)$:** $$P(A) = 1 - P(A') = 1 - \frac{3}{8} = \frac{5}{8}$$ 4. **Substitute known values into the total probability formula:** $$\frac{2}{5} = P(B|A) \times \frac{5}{8} + \frac{7}{10} \times \frac{3}{8}$$ 5. **Calculate the second term:** $$\frac{7}{10} \times \frac{3}{8} = \frac{21}{80}$$ 6. **Rewrite the equation:** $$\frac{2}{5} = P(B|A) \times \frac{5}{8} + \frac{21}{80}$$ 7. **Isolate $P(B|A)$ term:** $$P(B|A) \times \frac{5}{8} = \frac{2}{5} - \frac{21}{80}$$ 8. **Find common denominator and subtract:** $$\frac{2}{5} = \frac{32}{80}$$ $$\frac{32}{80} - \frac{21}{80} = \frac{11}{80}$$ 9. **So:** $$P(B|A) \times \frac{5}{8} = \frac{11}{80}$$ 10. **Divide both sides by $\frac{5}{8}$:** $$P(B|A) = \frac{\frac{11}{80}}{\frac{5}{8}} = \frac{11}{80} \times \frac{8}{5} = \frac{88}{400}$$ 11. **Simplify $\frac{88}{400}$:** $$\frac{88}{400} = \frac{\cancel{88}^{11}}{\cancel{400}^{50}} = \frac{11}{50}$$ 12. **Now find $P(A \cap B)$ using:** $$P(A \cap B) = P(B|A)P(A) = \frac{11}{50} \times \frac{5}{8}$$ 13. **Multiply:** $$\frac{11}{50} \times \frac{5}{8} = \frac{55}{400}$$ 14. **Simplify $\frac{55}{400}$:** $$\frac{55}{400} = \frac{\cancel{55}^{11}}{\cancel{400}^{80}} = \frac{11}{80}$$ **Final answer:** $$P(A \cap B) = \frac{11}{80}$$