Mathematics 10 — Exercise 12.1 · Q6 (iii b)

Full question

Two fair six-sided dice are rolled. Find the probability that the product of the two numbers is prime.

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.

   $$\mathrm{n(E)}=6$$

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

   $$\frac{6}{36}$$

4. Calculate this operation before combining the remaining terms.

   $$6\times 1=6$$

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.

   $$\frac{6\times 1}{1\times 36}$$

7. Calculate these two products separately.

   $$\frac{6}{36}$$

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

   $$\mathrm{GCD}=6$$

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

   $$\frac{6\times 1}{1\times 6}$$

10. Calculate these two products separately.

    $$\frac{6}{6}$$

11. Show the two divisions explicitly.

    $$\frac{\frac{6}{6}}{\frac{6}{6}}$$

12. Write the reduced fraction.

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

13. Calculate this operation before combining the remaining terms.

    $$6\times \frac{1}{6}=1$$

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

    $$\frac{36\times 1}{1\times 6}$$

15. Calculate these two products separately.

    $$\frac{36}{6}$$

16. Show the two divisions explicitly.

    $$\frac{\frac{36}{6}}{\frac{6}{6}}$$

17. Write the reduced fraction.

    $$6\times \frac{1}{1}$$

18. Calculate this operation before combining the remaining terms.

    $$36\times \frac{1}{6}=6$$

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

    $$\frac{1\times 1}{1\times 6}$$

20. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{6}=\frac{1}{6}$$

21. Show the two divisions explicitly.

    $$\frac{\frac{6}{6}}{\frac{36}{6}}$$

22. Evaluate the numerator division.

    $$\frac{6}{6}=1$$

23. Evaluate the denominator division.

    $$\frac{36}{6}=6$$

24. Write the reduced fraction.

    $$\frac{1}{6}$$

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.

    $$6\times \frac{1}{36}=\frac{1}{6}$$

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

    $$\frac{1}{6}$$

Answer

The final answer is shown below.