Mathematics 10 — Exercise 12.2 · Q1

Full question

P(A)=1/4,P(B)=1/2,P(A∩B)=1/8. Find P(A∪B).

Step-by-step solution

1. For an inclusive OR, add the two probabilities and subtract their overlap once: P(A∪B)=P(A)+P(B)−P(A∩B).

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

   $$\frac{1\times 1}{1\times 4}$$

3. Calculate these two products separately.

   $$\frac{1}{4}$$

4. Calculate this operation before combining the remaining terms.

   $$1\times \frac{1}{4}=\frac{1}{4}$$

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

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

6. Calculate these two products separately.

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

7. Calculate this operation before combining the remaining terms.

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

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

   $$\frac{1\times 1}{1\times 8}$$

9. Calculate these two products separately.

   $$\frac{1}{8}$$

10. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{8}=\frac{1}{8}$$

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

    $$\frac{\left(-1\right)\times 1}{1\times 8}$$

12. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{8}=-\frac{1}{8}$$

13. Substitute the given probabilities or outcome counts.

    $$\frac{1}{4}+\frac{1}{2}-\frac{1}{8}$$

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

    $$\mathrm{LCD}=8$$

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

    $$\frac{1}{4}=\frac{1\times 2}{4\times 2}$$

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

    $$\frac{2}{8}$$

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

    $$\frac{1}{2}=\frac{1\times 4}{2\times 4}$$

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

    $$\frac{4}{8}$$

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

    $$-\frac{1}{8}=\frac{\left(-1\right)\times 1}{8\times 1}$$

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

    $$-\frac{1}{8}$$

21. Calculate this operation before combining the remaining terms.

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

22. Calculate this operation before combining the remaining terms.

    $$2+4-1=5$$

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

    $$\frac{5\times 1}{1\times 8}$$

24. Calculate these two products separately.

    $$\frac{5}{8}$$

25. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{8}=\frac{5}{8}$$

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

    $$\frac{2+4-1}{8}=\frac{5}{8}$$

27. Calculate this operation before combining the remaining terms.

    $$\frac{1}{4}+\frac{1}{2}-\frac{1}{8}=\frac{5}{8}$$

28. The simplified result is the required probability.

    $$\frac{5}{8}$$

Answer

The final answer is shown below.