The sum of the number is even when two dice rolled?

Answer 1

One way to do this is to list all the possible outcomes. You can get an even sum with two even numbers or two odd numbers. So here is the list.

(1,1) (3,1)

(5,1) (2,2)

(4,2) (6,2)

(1,3) (3,3)

(5,3) (2,4)

(4,4) (6,4)

(1,5) (3,5)

(5,5) (2,6)

(4,6) (6,6)

That is 18/36 so half the time the sum will be even. Second method to calculate the probability:

Also, note that to obtain an odd sum, we need and odd number one one die and an even number on the other one. So let's use that and figure out the answer another way.

The chance of even on the first is 1/2 and odd on the second is 1/2 so the chance of even and then odd is 1/4. Now, we can also have odd on the first roll then even. That chance is also 1/4. We add these two to get 1/2 which is what we found from enumeration of the possible outcomes. This is a good example of when to add and when to multiply probabilities. When we want both and odd and an even roll, we multiply because we want two events to occur at the same time and the probability of that is less than either one by itself. This is the multiplication principle.

However, when we want to look at the probability of EITHER and odd roll followed by an even or and even followed by an odd, we must ADD. Clearly these two outcomes can not occur at the same time since the first roll must be odd or even and can't be both. This tells us the chance of an odd sum is 1/2 which means the chance of an even sum is 1-1/2 or 1/2 also. Method three:

To obtain an even sum we need either two evens or two odds. Using the logic above in method 2, we see the probability of two evens is 1/4 and of two odd is 1/4. So the probability of either one of these is 1/4+1/4=1/2