Question

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

Hint:

Using the complement rule \(P(E') = 1 - P(E)\) is a powerful way to simplify probability calculations.
For product-related dice problems, calculating "at least one even" directly would require listing three cases (Even-Odd, Odd-Even, Even-Even), whereas calculating "both odd" takes only one simple step!

Answer 1

Step 1: Understanding the Question:
The problem asks for the probability of two events when two distinct dice are rolled simultaneously:
1. The sum of the numbers on the two dice is an even number.
2. The product of the numbers on the two dice is an even number.

Step 2: Key Formula or Approach:
1. When two six-sided dice are thrown, the total number of possible outcomes in the sample space is:
\[ N = 6 \times 6 = 36 \]
2. The probability of an event \(E\) is:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]

Step 3: Detailed Explanation:
1. Part (i): Even Sum
- The sum of two numbers is even if both numbers are even, or if both numbers are odd.
- Let \(x\) and \(y\) be the numbers obtained on the first and second die.
- Odd numbers on a die: \(\{1, 3, 5\}\) (3 choices).
- Even numbers on a die: \(\{2, 4, 6\}\) (3 choices).
- Case 1: Both numbers are odd. Number of outcomes = \(3 \times 3 = 9\).
- Case 2: Both numbers are even. Number of outcomes = \(3 \times 3 = 9\).
- Total favorable outcomes for an even sum = \(9 + 9 = 18\).
- Calculate the probability:
\[ P(\text{Even Sum}) = \frac{18}{36} = \frac{1}{2} \]
2. Part (ii): Even Product
- The product of two numbers is even if at least one of the numbers is even.
- The product is odd only if both numbers are odd.
- It is easier to use the complement rule:
\[ P(\text{Even Product}) = 1 - P(\text{Odd Product}) \]
- A product is odd if both numbers are odd. As calculated in Case 1, the number of outcomes with both odd is 9.
- Therefore, the probability of obtaining an odd product is:
\[ P(\text{Odd Product}) = \frac{9}{36} = \frac{1}{4} \]
- Calculate the probability of an even product:
\[ P(\text{Even Product}) = 1 - \frac{1}{4} = \frac{3}{4} \]

Step 4: Final Answer:
The probability of obtaining an even sum is \(\frac{1}{2}\) and the probability of obtaining an even product is \(\frac{3}{4}\).