Mathematics 10 — Exercise 12.3 · Q6 (iii)

Full question

Two cards are drawn successively at random from a standard deck of 52 cards, without replacement. Find the probability that both cards are spades.

Step-by-step solution

1. The first draw has 13 favourable cards or balls out of 52.

   $$\frac{13}{52}$$

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.

   $$\frac{12}{51}$$

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

   $$\frac{13\times 1}{1\times 52}$$

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

   $$\mathrm{GCD}=13$$

5. Show the two divisions explicitly.

   $$\frac{\frac{13}{13}}{\frac{52}{13}}$$

6. Evaluate the numerator division.

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

7. Evaluate the denominator division.

   $$\frac{52}{13}=4$$

8. Write the reduced fraction.

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

9. Calculate this operation before combining the remaining terms.

   $$13\times \frac{1}{52}=\frac{1}{4}$$

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

    $$\frac{12\times 1}{1\times 51}$$

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

    $$\mathrm{GCD}=3$$

12. Show the two divisions explicitly.

    $$\frac{\frac{12}{3}}{\frac{51}{3}}$$

13. Evaluate the numerator division.

    $$\frac{12}{3}=4$$

14. Evaluate the denominator division.

    $$\frac{51}{3}=17$$

15. Write the reduced fraction.

    $$\frac{4}{17}$$

16. Calculate this operation before combining the remaining terms.

    $$12\times \frac{1}{51}=\frac{4}{17}$$

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

    $$\frac{1\times 4}{4\times 17}$$

18. Calculate these two products separately.

    $$\frac{4}{68}$$

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

    $$\mathrm{GCD}=4$$

20. Show the two divisions explicitly.

    $$\frac{\frac{4}{4}}{\frac{68}{4}}$$

21. Evaluate the numerator division.

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

22. Evaluate the denominator division.

    $$\frac{68}{4}=17$$

23. Calculate this operation before combining the remaining terms.

    $$\frac{1}{4}\times \frac{4}{17}=\frac{1}{17}$$

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

    $$\frac{13}{52}\times \frac{12}{51}$$

25. Calculate this operation before combining the remaining terms.

    $$13\times 12=156$$

26. Calculate this operation before combining the remaining terms.

    $$52\times 51=2652$$

27. Multiply the numerators and multiply the denominators separately.

    $$\frac{13\times 12}{52\times 51}$$

28. Calculate this operation before combining the remaining terms.

    $$156\times 1=156$$

29. Calculate this operation before combining the remaining terms.

    $$1\times 2652=2652$$

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

    $$\frac{156\times 1}{1\times 2652}$$

31. Calculate these two products separately.

    $$\frac{156}{2652}$$

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

    $$\mathrm{GCD}=156$$

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

    $$\frac{156\times 1}{1\times 156}$$

34. Calculate these two products separately.

    $$\frac{156}{156}$$

35. Show the two divisions explicitly.

    $$\frac{\frac{156}{156}}{\frac{156}{156}}$$

36. Write the reduced fraction.

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

37. Calculate this operation before combining the remaining terms.

    $$156\times \frac{1}{156}=1$$

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

    $$\frac{2652\times 1}{1\times 156}$$

39. Calculate these two products separately.

    $$\frac{2652}{156}$$

40. Show the two divisions explicitly.

    $$\frac{\frac{2652}{156}}{\frac{156}{156}}$$

41. Write the reduced fraction.

    $$17\times \frac{1}{1}$$

42. Calculate this operation before combining the remaining terms.

    $$2652\times \frac{1}{156}=17$$

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

    $$\frac{1\times 1}{1\times 17}$$

44. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{17}=\frac{1}{17}$$

45. Show the two divisions explicitly.

    $$\frac{\frac{156}{156}}{\frac{2652}{156}}$$

46. Evaluate the numerator division.

    $$\frac{156}{156}=1$$

47. Evaluate the denominator division.

    $$\frac{2652}{156}=17$$

48. Write the reduced fraction.

    $$\frac{1}{17}$$

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

    $$\frac{1\times 1}{1\times 2652}$$

50. Calculate these two products separately.

    $$\frac{1}{2652}$$

51. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{2652}=\frac{1}{2652}$$

52. Calculate this operation before combining the remaining terms.

    $$156\times \frac{1}{2652}=\frac{1}{17}$$

53. Simplify the resulting fraction.

    $$\frac{1}{17}$$

Answer

The final answer is shown below.