The first things to know about probability and statistics are that everything occurs with a probabilty.  All events that occur in life have probabilities associated with them.  For our purposes we are going to only consider events with very clear probabilities.

Probabilities:

  What are probabilities?  Probabilities are simply how likely something is to occur, where 100% (1) probability is the highest and means that something is absolutely going to occur, and 0% (0) probability cannot occur.  Most of the time, probabilities are written using their decimal notation, instead of the percentage notation, so 75% is .75, 50% is .5, 100% is 1 and 0% is 0.  I will be using this notation throughout the rest of this material.
    Most of the time we take things in isolation, such as a flip of a coin we constrain to not be blown away, such that it never lands, such as someone letting go of it in space or other such instances.  So we highly constrain most events that we are concerned with to behaving within well defined terms.
    The simplest way to define the probability of something is to simply say that it is, the number of ways that something can occur, divided by the number of possible outcomes of a situation.

Examples:

1. For flipping a coin, there are only the outcomes Heads (H) or Tails (T), which if we count them, there are 2.  If we want to find the probability of a coin landing on its Head (H), then we have to divide the number of ways that outcome can occur ( { H } = 1 ) by the number of possible outcomes of the event ( { H, T } = 2 ).  So the probability is 1/2, or .5 (50%).
2. If we had a bag of 7 green balls, and 3 blue balls, and we wanted to know what the probability was that we would choose a blue ball, then we could follow a similar procedure.  How many ways can we pick a blue ball?  We can reach in and choose any of the 3 balls.  Though when reaching in, we could also choose a green ball.  So what is the probability of choosing the blue ball?  There are 3 blue balls in the bag, and therefore we can choose any of those three, meaning that there are 3 ways to get a blue ball.  Though there are 7 green balls in the bag, and therefore there are ten ways to get a green ball.  So if we add those, then there are 10 possible occurances, we could get a blue ball (3) + we could get a green ball (7).  Given that, since there are 3 blue balls, we have 3/10, which is .3.  So there is a 30% chance that we would grab a blue ball.1

Probabilities of Multiple Events: