Mathematics 10 — Exercise Review · Q4

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 red.

Step-by-step solution

1. The first draw has 26 favourable cards or balls out of 52.

   $$\frac{26}{52}$$

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.

   $$\frac{25}{51}$$

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

   $$\frac{26\times 1}{1\times 52}$$

4. Show the two divisions explicitly.

   $$\frac{\frac{26}{26}}{\frac{52}{26}}$$

5. Evaluate the numerator division.

   $$\frac{26}{26}=1$$

6. Evaluate the denominator division.

   $$\frac{52}{26}=2$$

7. Write the reduced fraction.

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

8. Calculate this operation before combining the remaining terms.

   $$26\times \frac{1}{52}=\frac{1}{2}$$

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

   $$\frac{25\times 1}{1\times 51}$$

10. Calculate this operation before combining the remaining terms.

    $$25\times \frac{1}{51}=\frac{25}{51}$$

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

    $$\frac{1\times 25}{2\times 51}$$

12. Calculate this operation before combining the remaining terms.

    $$\frac{1}{2}\times \frac{25}{51}=\frac{25}{102}$$

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

    $$\frac{26}{52}\times \frac{25}{51}$$

14. Calculate this operation before combining the remaining terms.

    $$26\times 25=650$$

15. Calculate this operation before combining the remaining terms.

    $$52\times 51=2652$$

16. Multiply the numerators and multiply the denominators separately.

    $$\frac{26\times 25}{52\times 51}$$

17. Calculate this operation before combining the remaining terms.

    $$650\times 1=650$$

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.

    $$\frac{650\times 1}{1\times 2652}$$

20. Calculate these two products separately.

    $$\frac{650}{2652}$$

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

    $$\mathrm{GCD}=26$$

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

    $$\frac{650\times 1}{1\times 26}$$

23. Calculate these two products separately.

    $$\frac{650}{26}$$

24. Show the two divisions explicitly.

    $$\frac{\frac{650}{26}}{\frac{26}{26}}$$

25. Write the reduced fraction.

    $$25\times \frac{1}{1}$$

26. Calculate this operation before combining the remaining terms.

    $$650\times \frac{1}{26}=25$$

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

    $$\frac{2652\times 1}{1\times 26}$$

28. Calculate these two products separately.

    $$\frac{2652}{26}$$

29. Show the two divisions explicitly.

    $$\frac{\frac{2652}{26}}{\frac{26}{26}}$$

30. Write the reduced fraction.

    $$102\times \frac{1}{1}$$

31. Calculate this operation before combining the remaining terms.

    $$2652\times \frac{1}{26}=102$$

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

    $$\frac{25\times 1}{1\times 102}$$

33. Calculate this operation before combining the remaining terms.

    $$25\times \frac{1}{102}=\frac{25}{102}$$

34. Show the two divisions explicitly.

    $$\frac{\frac{650}{26}}{\frac{2652}{26}}$$

35. Evaluate the numerator division.

    $$\frac{650}{26}=25$$

36. Evaluate the denominator division.

    $$\frac{2652}{26}=102$$

37. Write the reduced fraction.

    $$\frac{25}{102}$$

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.

    $$650\times \frac{1}{2652}=\frac{25}{102}$$

42. Simplify the resulting fraction.

    $$\frac{25}{102}$$

Answer

The final answer is shown below.