Probability of intersection in conditional probability Suppose there are seven keys:


5 red keys.
2 green keys.


Only one of those seven keys opens a lock.

If for the experiment "Choose a random key and see if it opens the lock", I define the following events:


A: "The chosen key opens the lock".
G: "The right key is green".


And knowing that:


The probability of A is: 1/7.
The probability of G is: 2/7.
The probability of A knowing that the key is green is: 1/2


According to the relationship in conditional probability, the probability of A intersection G is: 1/7

Because p(A|G) = A intersection G / p(G)

But according to the probability of an intersection of independent events, the probability of A intersection G should be: 1/7 * 2/7 = 2/49

I do not understand why I get different values.
This is not quite a random experiment as it is determined in advance which key opens the lock. So, your calculations involving G are illegal.
The problem is you are assuming in your last calculation that $A$ and $G$ are independent.  In general,
$$\mathbb{P}(A \cap G) = \mathbb{P}(A \mid G) \cdot \mathbb{P}(G) \neq \mathbb{P}(A) \cdot \mathbb{P}(G).$$
The fact that $\mathbb{P}(A \mid G) = \frac{1}{2}$ and not $\frac{1}{7}$ is telling you the two events are not independent, and that's the reason you're getting different values.