Step 1: Listing out possible outcomes
List out all possible outcomes of rolling two dice. There are 36 outcomes as there are 6 sides of each die.
Step 2: Identifying pairs with even sum
Identify and list out the pairs where the sum of the numbers is an even number. These are the favorable outcomes.
Step 3: Counting favorable outcomes
Count the number of pairs identified in Step 2. Let this count be represented by x.
Step 4: Calculating the probability
Calculate the probability of obtaining an even sum by dividing the number of favorable outcomes (x) by the total number of outcomes (36).
The probability of obtaining an even sum is given by:
\[P(Even\,sum) = \frac{x}{36}\]
Now, let's put this plan into action.
Step 1
Listing all the possible outcomes, we get a 6x6 grid (since there are 6 sides on each die):
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Step 2
We can find the pairs with an even sum:
(1,1) (1,3) (1,5) (2,2) (2,4) (2,6)
(3,1) (3,3) (3,5) (4,2) (4,4) (4,6)
(5,1) (5,3) (5,5) (6,2) (6,4) (6,6)
Step 3
Counting the pairs listed above, we have x = 18 favorable outcomes.
Step 4
Calculating the probability of obtaining an even sum:
\[P(Even\,sum) = \frac{x}{36} = \frac{18}{36} = \frac{1}{2}\]
Therefore, the probability of the sum of the numbers being an even number when rolling a pair of dice is \(\frac{1}{2}\).
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