Mathematics 10 — Exercise 12.3 · Q3

Full question

A fair six-sided die is rolled twice. Find the probability that one result is a multiple of 3 and the other result is 5.

Step-by-step solution

1. All outcomes are equally likely. Count the complete sample space.

   $$\mathrm{n(S)}=36$$

2. Select exactly the outcomes satisfying the written condition. These are listed below.

   $$\mathrm{n(E)}=4$$

3. Probability=favourable outcomes / total equally likely outcomes.

   $$\frac{4}{36}$$

4. Calculate this operation before combining the remaining terms.

   $$4\times 1=4$$

5. Calculate this operation before combining the remaining terms.

   $$1\times 36=36$$

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

   $$\frac{4\times 1}{1\times 36}$$

7. Calculate these two products separately.

   $$\frac{4}{36}$$

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

   $$\mathrm{GCD}=4$$

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

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

10. Calculate these two products separately.

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

11. Show the two divisions explicitly.

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

12. Write the reduced fraction.

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

13. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{36\times 1}{1\times 4}$$

15. Calculate these two products separately.

    $$\frac{36}{4}$$

16. Show the two divisions explicitly.

    $$\frac{\frac{36}{4}}{\frac{4}{4}}$$

17. Write the reduced fraction.

    $$9\times \frac{1}{1}$$

18. Calculate this operation before combining the remaining terms.

    $$36\times \frac{1}{4}=9$$

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

    $$\frac{1\times 1}{1\times 9}$$

20. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{9}=\frac{1}{9}$$

21. Show the two divisions explicitly.

    $$\frac{\frac{4}{4}}{\frac{36}{4}}$$

22. Evaluate the numerator division.

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

23. Evaluate the denominator division.

    $$\frac{36}{4}=9$$

24. Write the reduced fraction.

    $$\frac{1}{9}$$

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

    $$\frac{1\times 1}{1\times 36}$$

26. Calculate these two products separately.

    $$\frac{1}{36}$$

27. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{36}=\frac{1}{36}$$

28. Calculate this operation before combining the remaining terms.

    $$4\times \frac{1}{36}=\frac{1}{9}$$

29. Reduce the fraction by dividing numerator and denominator by their greatest common divisor.

    $$\frac{1}{9}$$

Answer

The final answer is shown below.