Yes, a fair coin toss will give us 50-50 for heads or tails independent of what has happened before. Thus one coin toss is independent of another.
Two unrelated individuals should get independent scores on a computer game independent of one another. If the individuals are related then the independence assumption may be violated.
Here is a video written for a stats and methods unit but it has an important real life example in it where ignoring the independence assumption lead to a conviction of Sally Clark.