TITLE: Are two events independent, if occurrence of one event alters possible outcomes of the other's, but still maintains the same probability?
QUESTION [7 upvotes]: Let's assume a group of $80$ friends where $40$ of them plays baseball, $20$ of them plays football and $10$ of them plays both the game. Without any further information, if a random person is chosen, the probability of him playing football is $20/80 = 1/4$
Now let's say, we introduce one more information that the chosen person plays baseball. Now the sample space is halved (currently having only $40$ people) but so is the event space (currently $10$). So probability stays the same ($1/4$)
This example was shown to me by one of my friends and he claimed, here One person playing baseball and One person playing football is independent of each other. Because the occurrence of one event is not changing other's probability.
But I thought, if one event reduces other's possible outcomes (Here $10$ friends leaves the equation when I apply condition), how can they be considered independent.
If these events are indeed independent, the same example with different numbers, would surely change the probability. Will the same example then be considered having dependent events?
The Wikipedia article on Conditional Probability says,

If P(A|B) = P(A), then events A and B are said to be independent: in
such a case, knowledge about either event does not alter the
likelihood of each other

Then should I always calculate the answer of any probability question to determine what formula to use? Then how will I get to the answer in the first place?
Edit
Fine tuned terminologies to match with what I wanted to say

REPLY [5 votes]: Good question!
When considering whether events $X$ and $Y$ are independent, the correct intuition is to ask whether the knowledge that $X$ happens changes the probability of $Y$—not whether it restricts how $Y$ can happen.
Consider this experiment: flip two fair coins, letting $H_1$ be the
event that the first coin lands on Heads, and $X$ be the event that
the coins land on different sides. Then $$ P\left(H_1 \cap
X\right)=\frac14=P(H_1)\:P(X);$$ i.e., Events $H_1$ and $X$ are, by definition, independent. Despite this, the knowledge that $H_1$ happens has reduced the number of ways that $X$ can happen: one of $X$'s outcomes, $TH,$ is no longer possible.
Summarising:

Let $P(A)\neq0.$
Then events $A$ and $B$ are independent iff. the knowledge that $A$ happens doesn't change $B$'s probability.

Visualising independence.