Mathematics 10 — Exercise Review · Q3

Full question

One card is drawn at random from a standard deck of 52 playing cards. Find the probability that the card is an ace or a spade.

Step-by-step solution

1. For an inclusive OR, add the two probabilities and subtract their overlap once: P(A∪B)=P(A)+P(B)−P(A∩B).

2. There are 4 aces,13 spades, and ONE ace of spades in the overlap.

3. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times 1=-1$$

4. Substitute the given probabilities or outcome counts.

   $$\frac{4+13-1}{52}$$

5. Calculate this operation before combining the remaining terms.

   $$4+13-1=16$$

6. Calculate this operation before combining the remaining terms.

   $$16\times 1=16$$

7. Calculate this operation before combining the remaining terms.

   $$1\times 52=52$$

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

   $$\frac{16\times 1}{1\times 52}$$

9. Calculate these two products separately.

   $$\frac{16}{52}$$

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

    $$\mathrm{GCD}=4$$

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

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

12. Calculate these two products separately.

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

13. Show the two divisions explicitly.

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

14. Evaluate the denominator division.

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

15. Write the reduced fraction.

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

16. Calculate this operation before combining the remaining terms.

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

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

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

18. Calculate these two products separately.

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

19. Show the two divisions explicitly.

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

20. Write the reduced fraction.

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

21. Calculate this operation before combining the remaining terms.

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

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

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

23. Calculate this operation before combining the remaining terms.

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

24. Show the two divisions explicitly.

    $$\frac{\frac{16}{4}}{\frac{52}{4}}$$

25. Evaluate the numerator division.

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

26. Evaluate the denominator division.

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

27. Write the reduced fraction.

    $$\frac{4}{13}$$

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

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

29. Calculate these two products separately.

    $$\frac{1}{52}$$

30. Calculate this operation before combining the remaining terms.

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

31. Calculate this operation before combining the remaining terms.

    $$16\times \frac{1}{52}=\frac{4}{13}$$

32. The simplified result is the required probability.

    $$\frac{4}{13}$$

Answer

The final answer is shown below.