# 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. **

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