Probability

In algebra, probability involves chances of at least one event either occurring or not occurring.

Basic
Basic probability is denoted as P(event) = x.

Here's how you calculate P(event):

$$P(event) = \frac{f}{p}\,$$, where f is the favorable outcome, where the event occurs, and p is the possible outcomes, where all the possible events occur.

Because 0 ≤ f ≤ p, P(event) can only be between 0 and 1, or 0% and 100%.

If an event has a 0 probability (or 0% chance) of occurring, then the event is impossible to occur.

If an event has a 1 probability (or 100% chance) of occurring, then the event will be absolutely guaranteed to occur.

Mutually Exclusive Events
When finding the probabilities of two events, one must take into account events that are mutually exclusive.

Events are mutually exclusive if they cannot occur simultaneously. (e.g., you cannot roll both a 1 and a 4 on one die!)

Events are mutually inexclusive if they can occur simultaneously. (e.g., you can roll a 1 and a 4 on two dice!)

Independent and Dependent Events
Independent events are events that don't influence each other. (e.g., rolling a die and flipping a coin are independent events)

Dependent events are events that influence each other. (e.g., the chance of getting hit by raindrops and the chances of a stormy day are dependent events)

Conditional Probabilities
If you run across P(x | y), you've ran across a conditional probability. Conditional probabilities limit your sample space. A sample space is a table where data is displayed and compared.

You read P(x | y) as, "The probability of x, given y."

Q: Does it make a difference the way you have x and y ordered in the parentheses?

A: Absolutely! Allow me to show you an example of why this is the case.

Suppose I take a survey of a classroom about if the students had a pet or not. Here is my sample space:

Let's see what switching our x (have pet?) and y (gender) does to the probabilities!

P(has pet | male) = 7 have pet / 11 total males → P(has pet | male) = 7/11.

P(male | has pet) = 7 males / 13 total pet owners → P(male | has pet) = 7/13.

Therefore, P(swapping x and y influencing probability) = 1!