Mathematics 10 — Exercise 12.2 · Q12

Full question

Events A and B are mutually exclusive. Given P(Aᶜ) = 0.45 and P(A ∪ B) = 0.65, find P(B).

Step-by-step solution

1. P(A)=1−0.45=0.55. With no overlap P(A∪B)=P(A)+P(B), so subtract P(A).

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

   $$\frac{13\times 1}{1\times 20}$$

3. Calculate this operation before combining the remaining terms.

   $$13\times \frac{1}{20}=\frac{13}{20}$$

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

   $$\frac{11\times 1}{1\times 20}$$

5. Calculate these two products separately.

   $$\frac{11}{20}$$

6. Calculate this operation before combining the remaining terms.

   $$11\times \frac{1}{20}=\frac{11}{20}$$

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

   $$\frac{\left(-1\right)\times 11}{1\times 20}$$

8. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times \frac{11}{20}=-\frac{11}{20}$$

9. Substitute the given probabilities or outcome counts.

   $$\frac{13}{20}-\frac{11}{20}$$

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

    $$\mathrm{LCD}=20$$

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

    $$\frac{13}{20}=\frac{13\times 1}{20\times 1}$$

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

    $$\frac{13}{20}$$

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

    $$\frac{\left(-11\right)\times 1}{1\times 20}$$

14. Calculate this operation before combining the remaining terms.

    $$\left(-11\right)\times \frac{1}{20}=-\frac{11}{20}$$

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

    $$-\frac{11}{20}=\frac{\left(-11\right)\times 1}{20\times 1}$$

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

    $$-\frac{11}{20}$$

17. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 11=-11$$

18. Calculate this operation before combining the remaining terms.

    $$13-11=2$$

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

    $$\frac{2\times 1}{1\times 20}$$

20. Calculate these two products separately.

    $$\frac{2}{20}$$

21. Calculate this operation before combining the remaining terms.

    $$2\times \frac{1}{20}=\frac{1}{10}$$

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

    $$\frac{13-11}{20}=\frac{2}{20}$$

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

    $$\mathrm{GCD}=2$$

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

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

25. Calculate these two products separately.

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

26. Show the two divisions explicitly.

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

27. Write the reduced fraction.

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

28. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{20\times 1}{1\times 2}$$

30. Calculate these two products separately.

    $$\frac{20}{2}$$

31. Show the two divisions explicitly.

    $$\frac{\frac{20}{2}}{\frac{2}{2}}$$

32. Write the reduced fraction.

    $$10\times \frac{1}{1}$$

33. Calculate this operation before combining the remaining terms.

    $$20\times \frac{1}{2}=10$$

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

    $$\frac{1\times 1}{1\times 10}$$

35. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{10}=\frac{1}{10}$$

36. Show the two divisions explicitly.

    $$\frac{\frac{2}{2}}{\frac{20}{2}}$$

37. Evaluate the numerator division.

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

38. Evaluate the denominator division.

    $$\frac{20}{2}=10$$

39. Write the reduced fraction.

    $$\frac{1}{10}$$

40. Calculate this operation before combining the remaining terms.

    $$\frac{13}{20}-\frac{11}{20}=\frac{1}{10}$$

41. The simplified result is the required probability.

    $$\frac{1}{10}$$

Answer

The final answer is shown below.