Mathematics 10 — Exercise Review · Q6

Full question

A bag contains 7 orange and 5 purple marbles. Two marbles are drawn successively at random without replacement. Find the probability that both are orange.

Step-by-step solution

1. The first draw has 7 favourable cards or balls out of 12.

   $$\frac{7}{12}$$

2. The first object is not replaced: the total drops by1. Reduce the required colour/rank count too ONLY if the first object belonged to that category.

   $$\frac{6}{11}$$

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

   $$\frac{7\times 1}{1\times 12}$$

4. Calculate this operation before combining the remaining terms.

   $$7\times \frac{1}{12}=\frac{7}{12}$$

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

   $$\frac{6\times 1}{1\times 11}$$

6. Calculate this operation before combining the remaining terms.

   $$6\times \frac{1}{11}=\frac{6}{11}$$

7. Calculate this operation before combining the remaining terms.

   $$\frac{7}{12}\times \frac{6}{11}=\frac{7}{22}$$

8. Multiply the probability of the first draw by the conditional probability of the second draw.

   $$\frac{7}{12}\times \frac{6}{11}$$

9. Calculate this operation before combining the remaining terms.

   $$7\times 6=42$$

10. Calculate this operation before combining the remaining terms.

    $$12\times 11=132$$

11. Multiply the numerators and multiply the denominators separately.

    $$\frac{7\times 6}{12\times 11}$$

12. Calculate this operation before combining the remaining terms.

    $$42\times 1=42$$

13. Calculate this operation before combining the remaining terms.

    $$1\times 132=132$$

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

    $$\frac{42\times 1}{1\times 132}$$

15. Calculate these two products separately.

    $$\frac{42}{132}$$

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

    $$\mathrm{GCD}=6$$

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

    $$\frac{42\times 1}{1\times 6}$$

18. Calculate these two products separately.

    $$\frac{42}{6}$$

19. Show the two divisions explicitly.

    $$\frac{\frac{42}{6}}{\frac{6}{6}}$$

20. Evaluate the denominator division.

    $$\frac{6}{6}=1$$

21. Write the reduced fraction.

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

22. Calculate this operation before combining the remaining terms.

    $$42\times \frac{1}{6}=7$$

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

    $$\frac{132\times 1}{1\times 6}$$

24. Calculate these two products separately.

    $$\frac{132}{6}$$

25. Show the two divisions explicitly.

    $$\frac{\frac{132}{6}}{\frac{6}{6}}$$

26. Write the reduced fraction.

    $$22\times \frac{1}{1}$$

27. Calculate this operation before combining the remaining terms.

    $$132\times \frac{1}{6}=22$$

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

    $$\frac{7\times 1}{1\times 22}$$

29. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{22}=\frac{7}{22}$$

30. Show the two divisions explicitly.

    $$\frac{\frac{42}{6}}{\frac{132}{6}}$$

31. Evaluate the numerator division.

    $$\frac{42}{6}=7$$

32. Evaluate the denominator division.

    $$\frac{132}{6}=22$$

33. Write the reduced fraction.

    $$\frac{7}{22}$$

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

    $$\frac{1\times 1}{1\times 132}$$

35. Calculate these two products separately.

    $$\frac{1}{132}$$

36. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{132}=\frac{1}{132}$$

37. Calculate this operation before combining the remaining terms.

    $$42\times \frac{1}{132}=\frac{7}{22}$$

38. Simplify the resulting fraction.

    $$\frac{7}{22}$$

Answer

The final answer is shown below.