#### Question: Two different dice are thrown together. Find the probability that the numbers obtained have
- even sum, and
- even product

#### Video Explanation

#### Explanatory Answer

##### (i) The sum of the numbers is even.

Sum of two numbers is even if

(a) both numbers are even (or)

(b) both numbers are odd.

**(a) Both even:** 3 possibilities for each die to turn out even i.e., 2, 4, and 6.

3 possibilities for one die and 3 for the other i.e., 3 × 3 = 9 outcomes where both are even.

The outcomes are (2, 2)(2, 4)(2, 6)(4, 2)(4, 4)(4, 6)(6, 2)(6, 4)(6, 6)

**(b) Both odd:** 3 possibilities for each die to turn out odd i.e., 1, 3, and 5.

3 possibilities for one die and 3 for the other i.e., 3 × 3 = 9 outcomes where both are odd.

The outcomes are (1, 1)(1, 3)(1, 5)(3, 1)(3, 3)(3, 5)(5, 1)(5, 3)(5, 5)

Total outcomes when sum is even = 9 + 9 = 18

Total outcomes when 2 different dice are thrown = 6 × 6 = 36

Probability that sum is even = \\frac{\text{number of outcomes when sum is even}}{\text{Total number of outcomes}}) = \\frac{18}{36}) = \\frac{1}{2})

##### (ii) Product is even when

(a) one of the dice turns out even and the other turns out odd.

(b) both are even.

**(a) One is even and the other is odd.**

3 even numbers. 3 odd numbers. First one odd and second even; first even and second odd = 3 × 3 × 2 = 18 outcomes

**The outcomes are**

(1, 2)(1, 4)(1, 6)(3, 2)(3, 4)(3, 6)(5, 2)(5, 4)(5, 6)

(2, 1)(2, 3)(2, 5)(4, 1)(4, 3)(4, 5)(6, 1)(6, 3)(6, 5)

**(b) Both are even.**

3 possibilities for the first and 3 for the second = 3 × 3 = 9 outcomes.

The outcomes are (2, 2)(2, 4)(2, 6)(4, 2)(4, 4)(4, 6)(6, 2)(6, 4)(6, 6) = 9 outcomes

Total outcomes when product is even = 18 + 9 = 27

Total outcomes when two different dice are thrown = 6 × 6 = 36

Probability that product is even = \\frac{\text{number of outcomes when product is even}}{\text{Total number of outcomes}}) = \\frac{27}{36}) = \\frac{3}{4})

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