Probability is a way of telling how likely it is that an event will happen. Probability can be written as a number between zero and one or as a percent between 0% and 100%. A probability of zero or 0% means something will never happen. Similarly, a probability of 1 or 100% means something happens with certainty  it happens every time.
This introduction covers the following subjects:  For more advanced topics, check out the following: 
Before we start to calculate probabilities, first we need to figure out all the possible ways that something can happen. The list of all the possible ways is called the sample space.
For simple events, like flipping a single coin, the sample space is easy to figure out  a coin can only be heads or tails. For more complicated events, like rolling dice or picking cards, we need a tool to help us figure out all the possibilities. One powerful tool is the Fundamental Counting Principle.
Another way to determine the sample space is to use combinations and permutations  the number of ways you can arrange or group things. Making lists is one way we can do this, but there are easier methods we will learn.
For simple events where there are only two possible choices, we study binomial probability to determine the sample space. One application is the study of how many heads to expect if you are tossing five coins at the same time for 100 times.
One of the tools we use for binomial probabilities is Pascal's Triangle, named for work done by the French mathematician Blaise Pascal.
Probability is the chance that something will happen divided by all the possible ways the thing can happen. The shorthand way of writing "probability of an event" is P(event). Probability can be written as a fraction, a decimal, or a percent. Here are some simple examples:
fraction  decimal  percent  
never happens  0  0  0% 
happens half the time  ½  0.5  50% 
always happens  1  1.0  100% 
Now we will look at a simple example  flip one coin. What is the probability that you get a head or P(h)?
Remember that there are only 2 possibilities  either head or tail, but never both  and you are interested in getting a head.
probability of one head = P(h) =  1 head  =  ½ 
2 possibilities 
It gets more complicated if you have 2 coins. To make it easier to see, pretend you have a nickel and a dime. What is the probability that you get 2 heads or P(2h)? We know that this means we must have a head for the nickel and a head for the dime, but to finish the problem we need to know all the possibilities, so we will make a list:
nickel  dime  from the table we can see that there are four different possibilities, but only one way to get 2 heads 

head  head  
head  tail  probability of two heads = P(2h) = 1 way out of 4 = ¼  
tail  head  
tail  tail 
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