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# Probability and Statistics

# Important Elements

The probability of an event

- is a fraction:

- is non-negative (that is, 0 or greater). There's no such thing as a negative number of outcomes.

- cannot be more than 1. There can't be more favorable outcomes than there are possible outcomes, so bring the optimism down a notch.

- is 0 if there are no favorable outcomes; that is, if the event is impossible.

- is 1 if every outcome is favorable; that is, the event includes all possible outcomes.

- is a fraction between 0 and 1 (Yes, we know we're repeating ourselves.)

- Also, any fraction between 0 and 1 can be a probability.

### Sample Problem

What is the probability of rolling 7 with a standard 6-sided die?

There are no favorable outcomes. Not unless you're standing at the counter of a magic shop and testing out one of their trick dice. 7 isn't one of the possible outcomes when we roll an everyday, run-of-the-mill die. Therefore,

,

so the probability of rolling 7 is 0. The good news is that, as long as you are rolling only one die, you can't crap out.

If an event has a probability of 1, it means the event is absolutely, positively guaranteed to happen when you do the corresponding experiment (for example, rolling a number less than 7 on a die). If an event has a probability of 0, that event can absolutely, positively *not* happen when you do the experiment (for example, rolling 7 on a die). For another example, the television show *The Event* has a 0 probability of winning an Emmy this year, as it is no longer on the air. Voters tend to show their support for programs that weren't so bad they got canceled.

We recommend keeping this in mind when doing problems. If you're asked the probability of an event you *know* can't happen, you know the probability is 0, so you don't need to worry about counting favorable outcomes. If you're asked the probability of an event that *has* to happen, you know the probability is 1, so you don't need to worry about counting favorable outcomes. For example, you know your dad will wake you up tomorrow morning using that obnoxious Donald Duck voice, so the probability is automatically 1. When possible, do things the easy way.