Mathematics 10 — Exercise 12.3 · Q7

Full question

Saleem draws two cards successively at random from a standard deck of 52 cards without replacement. Find the probability that the first card is a jack and the second is a queen.

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.

   $$\frac{4}{51}$$

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

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

4. Show the two divisions explicitly.

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

5. Evaluate the numerator division.

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

6. Evaluate the denominator division.

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

7. Write the reduced fraction.

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

8. Calculate this operation before combining the remaining terms.

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

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

   $$\frac{4\times 1}{1\times 51}$$

10. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{1\times 4}{13\times 51}$$

12. Calculate this operation before combining the remaining terms.

    $$\frac{1}{13}\times \frac{4}{51}=\frac{4}{663}$$

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

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

14. Calculate this operation before combining the remaining terms.

    $$4\times 4=16$$

15. Calculate this operation before combining the remaining terms.

    $$52\times 51=2652$$

16. Multiply the numerators and multiply the denominators separately.

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

17. Calculate this operation before combining the remaining terms.

    $$16\times 1=16$$

18. Calculate this operation before combining the remaining terms.

    $$1\times 2652=2652$$

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

    $$\frac{16\times 1}{1\times 2652}$$

20. Calculate these two products separately.

    $$\frac{16}{2652}$$

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

    $$\mathrm{GCD}=4$$

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

    $$\frac{16\times 1}{1\times 4}$$

23. Calculate these two products separately.

    $$\frac{16}{4}$$

24. Show the two divisions explicitly.

    $$\frac{\frac{16}{4}}{\frac{4}{4}}$$

25. Write the reduced fraction.

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

26. Calculate this operation before combining the remaining terms.

    $$16\times \frac{1}{4}=4$$

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

    $$\frac{2652\times 1}{1\times 4}$$

28. Calculate these two products separately.

    $$\frac{2652}{4}$$

29. Show the two divisions explicitly.

    $$\frac{\frac{2652}{4}}{\frac{4}{4}}$$

30. Write the reduced fraction.

    $$663\times \frac{1}{1}$$

31. Calculate this operation before combining the remaining terms.

    $$2652\times \frac{1}{4}=663$$

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

    $$\frac{4\times 1}{1\times 663}$$

33. Calculate this operation before combining the remaining terms.

    $$4\times \frac{1}{663}=\frac{4}{663}$$

34. Show the two divisions explicitly.

    $$\frac{\frac{16}{4}}{\frac{2652}{4}}$$

35. Evaluate the numerator division.

    $$\frac{16}{4}=4$$

36. Evaluate the denominator division.

    $$\frac{2652}{4}=663$$

37. Write the reduced fraction.

    $$\frac{4}{663}$$

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

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

39. Calculate these two products separately.

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

40. Calculate this operation before combining the remaining terms.

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

41. Calculate this operation before combining the remaining terms.

    $$16\times \frac{1}{2652}=\frac{4}{663}$$

42. Simplify the resulting fraction.

    $$\frac{4}{663}$$

Answer

The final answer is shown below.