Mathematics 10 — Exercise 12.3 · Q1 (ii)

Full question

Two numbers are selected at random from 1 to 10, with replacement after the first selection. Find the probability that their product is even.

Step-by-step solution

All outcomes are equally likely. Count the complete sample space.
$n(S) = 100$
Select exactly the outcomes satisfying the written condition. These are listed below.
$n(E) = 75$
Probability=favourable outcomes / total equally likely outcomes.
$\frac{75}{100}$
Calculate this operation before combining the remaining terms.
$75 \times 1 = 75$
Calculate this operation before combining the remaining terms.
$1 \times 100 = 100$
For multiplication of fractions, multiply the numerators together and the denominators together.
$\frac{75 \times 1}{1 \times 100}$
Calculate these two products separately.
$\frac{75}{100}$
The numerator and denominator have this greatest common divisor. Divide BOTH by it.
$GCD = 25$
For multiplication of fractions, multiply the numerators together and the denominators together.
$\frac{75 \times 1}{1 \times 25}$
Calculate these two products separately.
$\frac{75}{25}$
Show the two divisions explicitly.
$\frac{\frac{75}{25}}{\frac{25}{25}}$
Evaluate the denominator division.
$\frac{25}{25} = 1$
Write the reduced fraction.
$3 \times \frac{1}{1}$
Calculate this operation before combining the remaining terms.
$75 \times \frac{1}{25} = 3$
For multiplication of fractions, multiply the numerators together and the denominators together.
$\frac{100 \times 1}{1 \times 25}$
Calculate these two products separately.
$\frac{100}{25}$
Show the two divisions explicitly.
$\frac{\frac{100}{25}}{\frac{25}{25}}$
Write the reduced fraction.
$4 \times \frac{1}{1}$
Calculate this operation before combining the remaining terms.
$100 \times \frac{1}{25} = 4$
For multiplication of fractions, multiply the numerators together and the denominators together.
$\frac{3 \times 1}{1 \times 4}$
Calculate this operation before combining the remaining terms.
$3 \times \frac{1}{4} = \frac{3}{4}$
Show the two divisions explicitly.
$\frac{\frac{75}{25}}{\frac{100}{25}}$
Evaluate the numerator division.
$\frac{75}{25} = 3$
Evaluate the denominator division.
$\frac{100}{25} = 4$
Write the reduced fraction.
$\frac{3}{4}$
For multiplication of fractions, multiply the numerators together and the denominators together.
$\frac{1 \times 1}{1 \times 100}$
Calculate these two products separately.
$\frac{1}{100}$
Calculate this operation before combining the remaining terms.
$1 \times \frac{1}{100} = \frac{1}{100}$
Calculate this operation before combining the remaining terms.
$75 \times \frac{1}{100} = \frac{3}{4}$
Reduce the fraction by dividing numerator and denominator by their greatest common divisor.
$\frac{3}{4}$
A product is odd only when BOTH numbers are odd. Use 1−(5/10)²=3/4.

Favourable outcome
(1, 2)
(1, 4)
(1, 6)
(1, 8)
(1, 10)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(2, 7)
(2, 8)
(2, 9)
(2, 10)
(3, 2)
(3, 4)
(3, 6)
(3, 8)
(3, 10)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(4, 7)
(4, 8)
(4, 9)
(4, 10)
(5, 2)
(5, 4)
(5, 6)
(5, 8)
(5, 10)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
(6, 7)
(6, 8)
(6, 9)
(6, 10)
(7, 2)
(7, 4)
(7, 6)
(7, 8)
(7, 10)
(8, 1)
(8, 2)
(8, 3)
(8, 4)
(8, 5)
(8, 6)
(8, 7)
(8, 8)
(8, 9)
(8, 10)
(9, 2)
(9, 4)
(9, 6)
(9, 8)
(9, 10)
(10, 1)
(10, 2)
(10, 3)
(10, 4)
(10, 5)
(10, 6)
(10, 7)
(10, 8)
(10, 9)
(10, 10)

Answer

The final answer is shown below.

\frac{3}{4}