if two fair dice are rolled, what is the probablity that the total showing is even or less than seven?

less than 7 :

15 14 13 12 11
24 23 22 21
33 32 31
42 41
51
there are 15 of these

sum is even : 18 cases
but that includes 9 cases from the above less than 7 count.
total count = 18 + 15-9 = 24

prob(less than seven OR even sum) = 24/36 = 2/3

To find the probability of getting a total showing which is even or less than seven when rolling two fair dice, we can first determine the total number of possible outcomes. Then we count the number of outcomes that satisfy the condition of being even or less than seven. Finally, we divide the latter by the former to calculate the probability.

Step 1: Counting the total number of possible outcomes
When rolling two fair dice, each die can have six possible outcomes (numbers 1 to 6). Since there are two dice being rolled independently, the total number of outcomes is the product of their individual possibilities. So, the total number of possible outcomes is 6 * 6 = 36.

Step 2: Counting the favorable outcomes
To determine the favorable outcomes, where the total showing is either even or less than seven, we need to make a list of all possible combinations that satisfy this condition. Let's break it down:

- If the total showing is even, possible combinations are (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), and (6,6). This gives us a total of 18 combinations.

- If the total showing is less than seven, all possible combinations except (6,6) meet this condition. So, we have 36 - 1 = 35 combinations.

Step 3: Calculating the probability
Now that we have the count of favorable outcomes (53) and the count of total outcomes (36), we can calculate the probability using the formula:

Probability = Favorable outcomes / Total outcomes

Thus, the probability of rolling two fair dice and getting a total showing that is even or less than seven is 53 / 36, which can be simplified as 1.47 or approximately 0.778.

Therefore, the probability is approximately 0.778 or 77.8%.