In the previous section, we were able to recognize whether an event was impossible, guaranteed, likely, unlikely, or equally likely to happen or not happen. It would be nice, though, if we had a mathematical formula which would help us determine just how likely or unlikely something is to happen.

Here's the formula:

Probability = .

Desired outcomes: If we're being asked for the probability of something happening, "desired outcomes" is the number of ways that the "something" could happen.

Possible outcomes: Possible outcomes is the number of ways an event could happen, regardless of whether or not we get the "desired" outcome. We also refer to the "possible outcomes" as the "sample space."

It's pretty simple to use, so let's dive right into an example.

Example One 
I roll a six-sided die. What is the probability that I will roll a two?

Answer
How many ways can the six-sided die land? It can land in 6 different ways (1, 2, 3, 4, 5, or 6). Of these six, how many are "desired" outcomes (an outcome I want). There is only one desired outcome - a two. Thus, since there are 6 possible outcomes, and only 1 desired outcome:

P =

Example Two
I roll a six-sided die. What is the probability that I will roll a prime number?

Answer
There are still 6 possible outcomes, but now the number of desired outcomes has changed. Now we are hoping for either 2, 3, or 5 to appear, since those are all primes. How many desired outcomes? Three!

P = =

Note that this means we are just as likely to get a prime as to not get a prime.

Example Three
I take a card from a 52 card deck. What is the probability the card will be an ace?

Answer
There are 52 possible outcomes. Of those, there are 4 desired outcomes (ace of spades, ace of hearts, ace of diamonds, and ace of clubs). 

P = =

Example Four
I take a card from a 52 card deck. What is the probability that it is a 14?

Answer
There are 52 possible outcomes. There are no desired outcomes, since there are no fourteens in the deck.

P = = 0

1.

I rolled a six-sided die. What is the probability that the number showing is a perfect square?

2.

I rolled a four-sided die. What is the probability that the number showing is a seven?

3.

I pulled one card from a deck of cards. What is the probability that it is a 3 or a 7?

4.

I pulled one card from a deck of cards. What is the probability that it is a 5 or a spade?

5.

A hat contains slips of paper with all the integers from one to ten inclusive. If I randomly draw a number, what is the probability that it will be less than four?

6.

I pick a letter from the alphabet. What is the probability that it will be a vowel?

7.

I randomly pick a date in the future. What is the probability that it is a weekend date?

8.

I have a jar with 5 pennies, 4 dimes, and 3 quarters. If I randomly pick one coin, what is the probability that I'll have at least ten cents?

9.

I have a drawer with 3 blue socks, 2 red socks, and 4 green socks. If I pick one sock randomly, what is the probability that the sock will be either blue, red, or green?

10.

I am holding a blue sock and a green sock. In my drawer I have 4 red socks, 2 green socks, and 1 blue sock. If I pick one sock at random from the drawer, what is the probability that I'll have a matched pair of socks?