Mathematics 10 — Exercise 12.1 · Q9 (iv)

Full question

One integer is selected at random from the integers 1 to 15 inclusive, all with equal probability. Find the probability of selecting a prime number.

Step-by-step solution

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

   $$\mathrm{n(S)}=15$$

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.

   $$\frac{6}{15}$$

4. Calculate this operation before combining the remaining terms.

   $$6\times 1=6$$

5. Calculate this operation before combining the remaining terms.

   $$1\times 15=15$$

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

   $$\frac{6\times 1}{1\times 15}$$

7. Calculate these two products separately.

   $$\frac{6}{15}$$

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

   $$\mathrm{GCD}=3$$

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

   $$\frac{6\times 1}{1\times 3}$$

10. Calculate these two products separately.

    $$\frac{6}{3}$$

11. Show the two divisions explicitly.

    $$\frac{\frac{6}{3}}{\frac{3}{3}}$$

12. Evaluate the denominator division.

    $$\frac{3}{3}=1$$

13. Write the reduced fraction.

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

14. Calculate this operation before combining the remaining terms.

    $$6\times \frac{1}{3}=2$$

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

    $$\frac{15\times 1}{1\times 3}$$

16. Calculate these two products separately.

    $$\frac{15}{3}$$

17. Show the two divisions explicitly.

    $$\frac{\frac{15}{3}}{\frac{3}{3}}$$

18. Write the reduced fraction.

    $$5\times \frac{1}{1}$$

19. Calculate this operation before combining the remaining terms.

    $$15\times \frac{1}{3}=5$$

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

    $$\frac{2\times 1}{1\times 5}$$

21. Calculate this operation before combining the remaining terms.

    $$2\times \frac{1}{5}=\frac{2}{5}$$

22. Show the two divisions explicitly.

    $$\frac{\frac{6}{3}}{\frac{15}{3}}$$

23. Evaluate the numerator division.

    $$\frac{6}{3}=2$$

24. Evaluate the denominator division.

    $$\frac{15}{3}=5$$

25. Write the reduced fraction.

    $$\frac{2}{5}$$

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

    $$\frac{1\times 1}{1\times 15}$$

27. Calculate these two products separately.

    $$\frac{1}{15}$$

28. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{15}=\frac{1}{15}$$

29. Calculate this operation before combining the remaining terms.

    $$6\times \frac{1}{15}=\frac{2}{5}$$

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

    $$\frac{2}{5}$$

Answer

The final answer is shown below.