Some more examples of independence:

Here are some INDEPENDENT events:

• You flip a coin and get a heads and you flip a second coin and get a tails.
The two coins don't influence each other.
• The probability of rain today and the probability of a pop quiz;
Quizzes happen, rain or shine.

Here are some NON-INDEPENDENT events:

• You draw one card from a deck and its black and you draw a second card and its black.
By removing one black card, you made the probability of drawing a second one slightly smaller. Technically this is called 'sampling without replacement'.
• The probability of snow today and the probability of a pop quiz;
Snow causes school closings, in which case your teacher can't give you a pop quiz.
• The chance that you are hungry right now and the chance that you're eating right now
Obviously one leads to the other eventually ...

Showing that two events are truly independent events can be difficult, because you have to show that they don't influence each other at all. However, lots of events are mostly independent, and therefore we can treat them as independent.