Mathematics 10 — Exercise 12.2 · Q13

Full question

P(A)=2/3,P(B)=2/5,P(A∪B)=1/3. Find P(A∩B).

Step-by-step solution

1. Rearrange the addition law: P(A∩B)=P(A)+P(B)−P(A∪B).

2. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{2\times 1}{1\times 3}$$

3. Calculate these two products separately.

   $$\frac{2}{3}$$

4. Calculate this operation before combining the remaining terms.

   $$2\times \frac{1}{3}=\frac{2}{3}$$

5. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{2\times 1}{1\times 5}$$

6. Calculate these two products separately.

   $$\frac{2}{5}$$

7. Calculate this operation before combining the remaining terms.

   $$2\times \frac{1}{5}=\frac{2}{5}$$

8. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{1\times 1}{1\times 3}$$

9. Calculate these two products separately.

   $$\frac{1}{3}$$

10. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{3}=\frac{1}{3}$$

11. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 3}$$

12. Calculate these two products separately.

    $$-\frac{1}{3}$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{3}=-\frac{1}{3}$$

14. Substitute the given probabilities or outcome counts.

    $$\frac{2}{3}+\frac{2}{5}-\frac{1}{3}$$

15. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=15$$

16. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{2}{3}=\frac{2\times 5}{3\times 5}$$

17. Calculate the converted numerator; the denominator is now common.

    $$\frac{10}{15}$$

18. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{2}{5}=\frac{2\times 3}{5\times 3}$$

19. Calculate the converted numerator; the denominator is now common.

    $$\frac{6}{15}$$

20. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{1}{3}=\frac{\left(-1\right)\times 5}{3\times 5}$$

21. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-5\right)\times 1}{1\times 15}$$

22. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

23. Show the two divisions explicitly.

    $$\frac{-\frac{5}{5}}{\frac{15}{5}}$$

24. Evaluate the numerator division.

    $$-\frac{5}{5}=-1$$

25. Evaluate the denominator division.

    $$\frac{15}{5}=3$$

26. Calculate this operation before combining the remaining terms.

    $$\left(-5\right)\times \frac{1}{15}=-\frac{1}{3}$$

27. Calculate the converted numerator; the denominator is now common.

    $$-\frac{5}{15}$$

28. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 5=-5$$

29. Calculate this operation before combining the remaining terms.

    $$10+6-5=11$$

30. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{11\times 1}{1\times 15}$$

31. Calculate these two products separately.

    $$\frac{11}{15}$$

32. Calculate this operation before combining the remaining terms.

    $$11\times \frac{1}{15}=\frac{11}{15}$$

33. Keep the common denominator and add the signed numerators.

    $$\frac{10+6-5}{15}=\frac{11}{15}$$

34. Calculate this operation before combining the remaining terms.

    $$\frac{2}{3}+\frac{2}{5}-\frac{1}{3}=\frac{11}{15}$$

35. The simplified result is the required probability.

    $$\frac{11}{15}$$

36. The arithmetic produces 11/15, but an intersection cannot exceed P(B)=2/5=6/15. Also the union cannot be smaller than P(A). The printed givens are inconsistent.

Answer

No valid events satisfy the printed data. Formal rearrangement gives 11/15, an impossible intersection.