how do I Calculate the probability, six-sided Dice.?

These are the numbers divisible by 2:

{2, 4, 6}

These are the numbers divisible by 3:

{3, 6}

So if you blindly count the two sets and add them, you'll get 5 numbers. However, there's a number that is on both lists (6) which is divisible by both 2 and 3. You need to subtract that so it only appears once in the combined set.

{2, 3, 4, 6}

There are 4 numbers (out of 6) that are divisible by 2 or 3.

Divide to get the probability:

4/6

Reduce the fraction to lowest terms:

2/3

That's the intuitive way to solve this. Now let's look at a more mathematically formal way to solve it.

You might have learned this formula:

P(A or B) = P(A) + P(B) - P(A and B)

A : roll is divisible by 2 {2,4,6}

B : roll is divisible by 3 {3,6}

A and B : roll is divisible by 2 and 3 (aka is divisible by 6) {6}

P(A) = 3/6

P(B) = 2/6

P(A and B) = 1/6

P(A or B) = (3/6 + 2/6) - 1/6

= 5/6 - 1/6

= 4/6

= 2/3

There are six possible outcomes.

Four of them satisfy the criterion: 2, 3, 4, and 6.

The required probability is 4/6 = 2/3.