Q) Two different dice are thrown together. Find the probability that the numbers obtained have:
(i) even sum
(ii) even product.

(Q 30 – 30/2/3 – CBSE 2026 Question Paper)

Ans:

Step 1: We know that in case of n dice, the total outcomes = (6)ⁿ

∵ here we have 2 dice, ∴ total outcomes = (6)² = 36 outcomes

(i) Probability to get EVEN SUM: 

Step 2: A sum is even if both dice show even numbers or both show odd numbers.

Case 1: Even numbers on both dice:

∵ even numbers on a die = {2,4,6} i.e. 3 choices

∴ when both dice will show even numbers, possible outcomes = 3 × 3 = 9 outcomes

∴ Favourable outcomes by even numbers on both dice = 9 outcomes

Case 2: Odd numbers on both dice:

∵ odd numbers on a die = {1,3,5} i.e. 3 choices

∴ when both dice will show odd numbers, possible outcomes = 3 × 3 = 9 outcomes

∴ Favourable outcomes by odd numbers on both dice = 9 outcomes

∴ Total favourable outcomes by even numbers or odd numbers on both dice = 9 + 9 = 18 outcomes

Step 3: We know that the Probability is given by:

Probability to get even sum, P_{Even sum} =

Therefore, the probability to get even sum by throw of two dice =

(ii) Probability to get EVEN PRODUCT: 

Step 4: A product is even if at least one die shows an even number.(it means, either 1st die shows even number and 2nd die shows odd number, or 1st die shows odd number and 2nd die shows even number or both dice show even number)

∴ Either we calculate outcomes from each of these 3 cases

OR

we calculate outcomes of getting odd product (i.e. both dice show odd number) and subtract from total outcomes 

Step 5: Let’s start with odd product (easier option)

∵ odd numbers on a die = {1,3,5} i.e. 3 choices

∴ when both dice will show odd numbers, possible outcomes = 3 × 3 = 9 outcomes

These 9 outcomes will give odd product

∴ Total favourable outcomes to get odd product = 9  outcomes

∴ Total favourable outcomes to get even product = total outcomes – Outcomes for odd product

= 36 – 9 = 27 outcomes

Step 6:

∵ Probability = 

∴ Probability to get even product, P_{Even product} =

Therefore, the probability to get even product by throw of two dice =

Final Answer: The probability to get even sum is and the probability to get even product is .

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