Mathematics 10 — Exercise 12.1 · Q10

Full question

For an event A, P(A) = 7/10. Find P(Aᶜ), the probability that A does not occur.

Step-by-step solution

1. An event and its complement have total probability 1. Subtract the event probability from 1.

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

   $$\frac{7\times 1}{1\times 10}$$

3. Calculate these two products separately.

   $$\frac{7}{10}$$

4. Calculate this operation before combining the remaining terms.

   $$7\times \frac{1}{10}=\frac{7}{10}$$

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

   $$\frac{\left(-1\right)\times 7}{1\times 10}$$

6. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times \frac{7}{10}=-\frac{7}{10}$$

7. Substitute the given probabilities or outcome counts.

   $$1-\frac{7}{10}$$

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

   $$\mathrm{LCD}=10$$

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

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

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

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

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

    $$\frac{\left(-7\right)\times 1}{1\times 10}$$

12. Calculate this operation before combining the remaining terms.

    $$\left(-7\right)\times \frac{1}{10}=-\frac{7}{10}$$

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

    $$-\frac{7}{10}=\frac{\left(-7\right)\times 1}{10\times 1}$$

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

    $$-\frac{7}{10}$$

15. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 7=-7$$

16. Calculate this operation before combining the remaining terms.

    $$10-7=3$$

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

    $$\frac{3\times 1}{1\times 10}$$

18. Calculate these two products separately.

    $$\frac{3}{10}$$

19. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{10}=\frac{3}{10}$$

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

    $$\frac{10-7}{10}=\frac{3}{10}$$

21. Calculate this operation before combining the remaining terms.

    $$1-\frac{7}{10}=\frac{3}{10}$$

22. The simplified result is the required probability.

    $$\frac{3}{10}$$

Answer

The final answer is shown below.