Mathematics 10 — Exercise 12.1 · Q8 (ii)

Full question

A bag contains 4 red, 5 white, 6 green and 3 black balls. One ball is drawn at random. Find the probability that it is red.

Step-by-step solution

1. Count all balls for the denominator and all eligible colours for the numerator.

2. Substitute the given probabilities or outcome counts.

   $$\frac{4}{4+5+6+3}$$

3. Calculate this operation before combining the remaining terms.

   $$4+5+6+3=18$$

4. Calculate this operation before combining the remaining terms.

   $$4\times 1=4$$

5. Calculate this operation before combining the remaining terms.

   $$1\times 18=18$$

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

   $$\frac{4\times 1}{1\times 18}$$

7. Calculate these two products separately.

   $$\frac{4}{18}$$

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

   $$\mathrm{GCD}=2$$

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

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

10. Calculate these two products separately.

    $$\frac{4}{2}$$

11. Show the two divisions explicitly.

    $$\frac{\frac{4}{2}}{\frac{2}{2}}$$

12. Evaluate the denominator division.

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

13. Write the reduced fraction.

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

14. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{18\times 1}{1\times 2}$$

16. Calculate these two products separately.

    $$\frac{18}{2}$$

17. Show the two divisions explicitly.

    $$\frac{\frac{18}{2}}{\frac{2}{2}}$$

18. Write the reduced fraction.

    $$9\times \frac{1}{1}$$

19. Calculate this operation before combining the remaining terms.

    $$18\times \frac{1}{2}=9$$

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

    $$\frac{2\times 1}{1\times 9}$$

21. Calculate this operation before combining the remaining terms.

    $$2\times \frac{1}{9}=\frac{2}{9}$$

22. Show the two divisions explicitly.

    $$\frac{\frac{4}{2}}{\frac{18}{2}}$$

23. Evaluate the numerator division.

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

24. Evaluate the denominator division.

    $$\frac{18}{2}=9$$

25. Write the reduced fraction.

    $$\frac{2}{9}$$

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

    $$\frac{1\times 1}{1\times 18}$$

27. Calculate these two products separately.

    $$\frac{1}{18}$$

28. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{18}=\frac{1}{18}$$

29. Calculate this operation before combining the remaining terms.

    $$4\times \frac{1}{18}=\frac{2}{9}$$

30. The simplified result is the required probability.

    $$\frac{2}{9}$$

Answer

The final answer is shown below.