Mathematics 10 — Exercise 12.3 · Q6 (ii)

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 queens.

Step-by-step solution

1. The first draw has 4 favourable cards or balls out of 52.

   $$\frac{4}{52}$$

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.

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

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

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

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

   $$\mathrm{GCD}=4$$

5. Show the two divisions explicitly.

   $$\frac{\frac{4}{4}}{\frac{52}{4}}$$

6. Evaluate the numerator division.

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

7. Evaluate the denominator division.

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

8. Write the reduced fraction.

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

9. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{3\times 1}{1\times 51}$$

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

    $$\mathrm{GCD}=3$$

12. Show the two divisions explicitly.

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

13. Evaluate the numerator division.

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

14. Evaluate the denominator division.

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

15. Write the reduced fraction.

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

16. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{51}=\frac{1}{17}$$

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

    $$\frac{1\times 1}{13\times 17}$$

18. Calculate this operation before combining the remaining terms.

    $$\frac{1}{13}\times \frac{1}{17}=\frac{1}{221}$$

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

    $$\frac{4}{52}\times \frac{3}{51}$$

20. Calculate this operation before combining the remaining terms.

    $$4\times 3=12$$

21. Calculate this operation before combining the remaining terms.

    $$52\times 51=2652$$

22. Multiply the numerators and multiply the denominators separately.

    $$\frac{4\times 3}{52\times 51}$$

23. Calculate this operation before combining the remaining terms.

    $$12\times 1=12$$

24. Calculate this operation before combining the remaining terms.

    $$1\times 2652=2652$$

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

    $$\frac{12\times 1}{1\times 2652}$$

26. Calculate these two products separately.

    $$\frac{12}{2652}$$

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

    $$\mathrm{GCD}=12$$

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

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

29. Calculate these two products separately.

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

30. Show the two divisions explicitly.

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

31. Write the reduced fraction.

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

32. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{2652\times 1}{1\times 12}$$

34. Calculate these two products separately.

    $$\frac{2652}{12}$$

35. Show the two divisions explicitly.

    $$\frac{\frac{2652}{12}}{\frac{12}{12}}$$

36. Write the reduced fraction.

    $$221\times \frac{1}{1}$$

37. Calculate this operation before combining the remaining terms.

    $$2652\times \frac{1}{12}=221$$

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

    $$\frac{1\times 1}{1\times 221}$$

39. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{221}=\frac{1}{221}$$

40. Show the two divisions explicitly.

    $$\frac{\frac{12}{12}}{\frac{2652}{12}}$$

41. Evaluate the numerator division.

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

42. Evaluate the denominator division.

    $$\frac{2652}{12}=221$$

43. Write the reduced fraction.

    $$\frac{1}{221}$$

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

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

45. Calculate these two products separately.

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

46. Calculate this operation before combining the remaining terms.

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

47. Calculate this operation before combining the remaining terms.

    $$12\times \frac{1}{2652}=\frac{1}{221}$$

48. Simplify the resulting fraction.

    $$\frac{1}{221}$$

Answer

The final answer is shown below.